{/2‘ {V vh- ‘v‘ “'.c '\ I I V. :4; <5" . 'w. , Q"“‘-fi.£"‘ 2’9}.V.~,;\ ' ' "" ‘ . an. . . '11:.» j:. 1 n v ' ‘ , . a u "I: ~VI'5- nu. fly. . _ _‘ A I All. I l :( ‘ .3 “u,- .<._ .<_.. . ‘A‘fl-J sv- «'7 {-1944 If.“ . o . PJ-h¢’~t- ”Q 'M-b-«e-x A. . ‘ ,A. .. 11...,3: " 3:: , ml?) ' - \ . » ‘ .S-t-f-h_t - \>. .v . ”.5“. b fl. ;_,‘ R r a??? r}? i.“ .. ‘2‘}. g" ‘ K! .f. '3'. I! .mr. k 1. 2101 (a 50 5? LIBRARY ”1 Michigan State 1 University M IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII illll\llllllll‘l’lll“l'1"‘llllllllllNIUHI 1H 3 1293 00551 8851 f This is to certify that the dissertation entitled A THREE-Dmeuswum Fume ELEMENT ANALYSIS OF THE TEMPeRATuRE 9151121914110“ on 1115 Hook 01: A FARRowNG- House presented by Heeseumg Choi has been accepted towards fulfillment of the requirements for P]? D degree in Mnferfl‘fla Date flat. 3 1/958 MSU RETURNING MATERIALS: . Place in book drop to LJBRARJES . remove this checkout from “ your record. FINES will be charged if book is returned after the date stamped below. F388 ‘5 1935‘ A THREE-DINIENSIONAL FINITE ELENIENT ANALYSIS OF THE TENIPERATURE DISTRIBUTION ON THE FLOOR OF A FARROWING HOUSE By Heeseung Choi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Agricultural Engineering Department of Agricultural Engineering 1988 +4+x r” V‘ 5,) 544 ABS TRACT A THREE-DIMENSIONAL F INITE ELEMENT ANALYSIS OF THE TEMPERATURE DISTRIBUTION ON THE FLOOR OF A FARROVVING HOUSE By Heeseung Choi The typical hot water floor heating system for a solid floor farrowing house has a complicated pipe circuit that heats the pig creep area but avoids the area beneath the sow. The typical system is costly to construct, has a relatively high pumping resistance and provides a less-than-desirable temperature distribu- tion in the creep area. An improved hot water heating circuit that runs beneath the sow and the creep area has been used to eliminate some of the problems. This improved system uses insulation around the pipe and between the pipe and floor surface in the area beneath the sow to obtain the desired floor temperature in the SOW area. No design method exists for the improved system. The floor temperature provided by the improved pipe system is a function of the number, size, depth and spacing of the heat pipes, the insulation size and the placement, and the size of the fins that can be attached to the pipes. A three-dimensional finite element heat transfer program was used to calculate the temperature distribution on the floor surface for various arrangements of the new heating system. The finite ele ment method was also used to find a design condition for each of three possible Heeseung Choi arrangements. A three-dimensional finite element grid generation program was written specifically for this study to generate the large volume of input data required in a solution. Three different arrangements were studied: (1) three hot water pipes without fins, (2) three pipes with a steel fin attached, and (3) three pipes with a copper fin attached. Prototype designs that gave the most desirable temperature distribution on the floor were recommended for each case. The recommended heating systems provide six places in the creep area with the desired piglet tem- perature range and a sow area within the desired temperature range. The heat input of the sow to the floor was not incorporated into this study. The structural strength of the floor resulting from the placement of flat sheets of insulation in the concrete also was not investigated. The new designs reduce operation costs and the pumping energy requirements. The farrowing areas near the cooler end of the hot water pipe line can be heated to the desired temperature by adjusting the size of the fins attached to the pipes Approved Maj esso C Mg Approved [ M ‘ \' [A1 (4/ Department Chairman Copyright by HEESEUNG CHOI 1988 To my wife Yangjin and our son Sungwoo and my father and mother in Korea ACKNOWLED GNIENT The author wishes to express the deep gratitude and appreciation to his major professor, Dr. Larry J. Segerlind (Agricultural Engineering) for the gui- dance, knowledge and encouragement throughout the graduate program. To the other members of the guidance committee, Dr. Thomas H. Bur- khardt (Agricultural Engineering), Dr. Howard L. Person (Agricultural Engineer- ing), and Dr. Gary L. Cloud (Metallurgy, Mechanics and Material Science), the author owes a debt of their inspiration and advices. Gratitude is extented to Dr. John B. Gerrish (Agricultural Engineering) and Andy Thulin (Animal Science) for constructive suggestions and criticisms. Author also wish to thank Dr. John D. Carlson (Agricultural Engineering) for his encourage in Christianity. vi TABLE OF CONTENTS List of Tables ............................................................................................. ix List of Figures ............................................................................................ x Chapter Page I. INTRODUCTION ...................................................................................... 1 II. LITERATURE REVIEW ........................................................................... 4 2.1 Farrowing House ................................................................................. 4 2.2 Three-Dimensional Finite Element Analysis ....................................... 6 2.3 Finite Element Formulation ............................................................... 8 III. ANALYSIS OF A TYPICAL FARROVVING HOUSE ................................ 17 3.1 Farrowing Crate Dimensions and Finite Element Model .................... 17 3.2 Calculated Temperature Values .......................................................... 21 IV. CALCULATIONS RELATED TO THE DESIGN OF A HOT WATER HEATING SYSTEM ................................................................................. 26 4.1 Finite Element Grid Generation ......................................................... 27 4.2 Calculations Related to the Design of a Piping System ...................... 30 4.2.1 Placement depth of flat insulation (D) ................................... 32 4.2.2 Thickness of flat insulation (tF) ............................................. 32 4.2.3 Width of flat insulation (W) .................................................. 36 4.2.4 Thickness of perimeter insulation (tP) ................................... 36 4.2.5 Thickness of flat insulation above the perimeter insulated pipe ......................................................................................... 42 4.2.6 Width of flat insulation above the perimeter insulated pipe .. 42 vii 4.2.7 Summary ................................................................................ 49 d V. PROTOTYPE MODELS FOR HOT WATER HEATING SYSTEM IN A FARROWING HOUSE ..................................................................... 46 5.1 Three Heating Pipes without Fins ...................................................... 46 5.1.1 Thickness of flat insulation ............................................. 48 5.1.2 Width of flat insulation .................................................. 50 5.1.3 Length of perimeter insulation ........................................ 50 5.1.4 Recommended model ....................................................... 53 5.2 Three Heating Pipes with Steel Fins ................................................... 58 5.2.1 Length of fin ................................................................... 58 5.2.2 Thickness of fin ............................................................... 61 5.2.3 Recommended model ....................................................... 61 5.3 Three Heating Pipes with Copper Fins .............................................. 66 5.3.1 Length of fin ................................................................... 70 5.3.2 Thickness of fin ............................................................... 70 5.3.3 Width of flat insulation .................................................. 75 5.3.4 Recommended model ....................................................... 75 5.4 Eflect of Room and Hot Water Temperature ...................................... 81 VI. DISCUSSION AND SUMMARY ................................................................ 86 VII. CONCLUSIONS ........................................................................................ 89 APPENDDC A GRID GENERATION PROGRAM ................................... 91 APPENDDC B FINITE ELEMENT HEAT TRANSFER PROGRAM ...... 98 APPENDIX C PIPE HEAD LOSS ........................................................... 109 BIBLIOGRAPHY ....................................................................................... 112 viii LIST OF TABLES Table Page 2.1 Shape functions and derivatives for eight node hexahedron. .................. 13 ix Figure 1.1 1.2 2.1 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 LIST OF FIGURES Page Pipe arrangement in a typical hot water heating system of the farrowing house which has 10 pens .......................................................................... 3 Pipe arrangement in a modified hot water heating system ...................... 3 Location of eight nodes in natural and Cartesian coordinates. ................ 12 Typical cross - section of concrete floor heated with hot water. .............. 18 Typical crate dimensions. ........................................................................ 19 Temperature distribution on the floor of the typical farrowing house heated with hot water. ............................................................................ 22 Temperature contour on the floor of the typical farrowing house heated with hot water. ............................................................................ 24 Temperature distribution on the floor when 1.3 cm thick and 5.1 cm wide flat insulation is applied over the hot water pipes. ......................... 25 Element for the model of three heat pipes without fin and the location of coordinates (Dimension of z—axis is expanded) ..................................... 29 Variables for the test model. ................................................................... 31 Effect of D, the placement depth of flat insulation (tF = 1.3 cm, W = 15.2 cm) .................................................................. 33 Effect of tF, the thickness of flat insulation (D = 2.5 cm, W = 15.2 cm) ................................................................... 34 Temperature drop according to the thickness of flat insulation. ............. 35 Effect of W, the width of flat insulation (tF = 1.3 cm, D = 2.5 cm). 37 4.7 4.8 4.9 4.10 4.11 4.12 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 Effect of W, the width of flat insulation (tF = 2.5 cm, D = 2.5 cm). 38 Temperature drop according to the width of flat insulation (tF = 1.3 cm, D = 2.5 cm). ................................................................... 39 Temperature drop according to the width of flat insulation (tF = 2.5 cm, D = 2.5 cm). ................................................................... 40 Effect of the thickness of perimeter insulation (No fiat insulation) .......... 41 Effect of the thickness of flat insulation on the perimeter insulated pipe (D = 2.5 cm, W = 15.2 cm, tP = 1.0 cm). ........................................... 43 Effect of the width of flat insulation on the perimeter insulated pipe (tF = 1.3 cm, D = 2.5 cm, tP = 1.0 cm). ............................................. 44 Variables for the model of three pipes without fin. ................................. 47 Effect of the thickness of flat insulation (L1 = 50.8 cm, W, = W2 = 15.2 cm, L2 = 76.2 cm). ............................ 49 Effect of the width of flat insulation (L1 = L2 = 50.8 cm, W, = 15.2 cm, TI = 1.3 cm). ............................... 51 Effect of the length of perimeter insulation (L1 = 50.8 cm, W, = 15.2 cm, W2 = 5.1 cm, T1 = 0.64 cm). ............... 52 Recommended model for three pipes without fin ..................................... 54 Temperature distribution on the floor of recommended model for three pipes without fin. ........................................................................... 55 Temperature contour on the floor of the recommended model for three pipes without fin. ........................................................................... 56 Three-dimensional temperature distribution of recommended model for three pipes without fin. ...................................................................... 57 Variables of fin and insulation for model of three pipes with fin ............. 59 Efiect of the length of steel fin (LI = 76.2 cm, WI = 15.2 cm, TI = 1.3 cm WF = 25.4 cm, TF = 0.5 cm). ......................................... 60 xi 5.11 5.12 5.13 5.14 5.15 5.15 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 Temperature change on the steel fin ........................................................ 62 Efiect of the thickness of flat insulation on 22.9 cm long steel fin. .......... 63 Effect of the thickness of steel fins ........................................................... 64 Recommended model for three pipes with steel fins. ............................... 65 Temperature distribution on the floor of recommended model for three pipes with steel fins. ....................................................................... 67 Temperature contour on the floor of the recommended model for three pipes with steel fins. ....................................................................... 68' Three-dimensional temperature distribution of recommended model for three pipes with steel fins. ................................................................ 69 Efi'ect of the length of copper fin (L1 = 7 6.2 cm, WI = 15.2 cm, TI = 1.3 cm, WF = 25.4 cm, TF = 0.5 cm). ........................................ 71 Temperature change on the copper fin. ................................................... 72 Effect of the thickness of flat insulation on 15.2 cm long copper fin. ...... 73 Effect of the thickness of copper fin ......................................................... 74 Effect of the width of flat insulation on 22.9 cm long copper fin ............. 76 Recommended model for three pipes with copper fin. ... .......................... 77 Temperature distribution on the floor of recommended model for three pipes with copper fins. .................................................................... 78 Temperature contour on the floor of the recommended model for three pipes with copper fins. .................................................................... 79 Three-dimensional temperature distribution of recommended model for three pipes with copper fins. ............................................................. 80 Effect of the hot water temperature at the room temperature 15.6 ° C.... 82 Effect of the room temperature at the water temperature 60 ° C. ............ 83 Temperature distribution in condition of 13.9 ° C (57°F) room temperature and 62.8 ° C (145 ° F) water temperature. ............................ 85 xii I. INTRODUCTION The farrowing house must provide a comfortable environmental condition for the young piglets and the sows at the same time. The new-born piglet needs a temperature of 294°C to 32.2'C (85°F to 90'F) to protect it from chilling, because it is poorly endowed with hair, has a low amount of body fat and a thin skin. Since chilling is a major cause of death in baby pigs, the new born piglet must be kept warm enough to survive the first three days of life. On the con- trary, the sow prefers a temperature of 156°C to 18.3'C (60’F to 65°F) to optimize feed intake, milk production, and sow condition. Two separate thermal environments are needed in a relatively small region of a farrowing house. To provide the environment for the piglets, additional heat sources are added in the baby pig creep area. Suspended infra-red lamps and suspended elec- tric bar heaters are often used in solid floor systems while heat pads are used to provide a warm micro-climate for the baby pigs on slotted floors. The use of these devices allows the remainder of the room to be maintained at a condition better for the sow. A hot water floor heating system is also used in farrowing houses to provide extra heat for the litter without excessive heating of the entire building. The hot water pipe system shown in Figure 1.1 by a dotted line is the typical arrangement used in farrowing buildings. This arrangement has many elbows that increase the construction time and cost, the likelihood of leaks, and 2 the operating cost of a swine facility. This complicated pipe line has been simplified to the one shown in Figure 1.2 where the heating pipes that cross the sow area are insulated to provide the proper temperature for the sow. The amount of insulation needed beneath the sow is not known. There are many variables that affect the temperature distribution on the floor of a farrowing house. Some of these include the number, spacing and depth of the pipes, the insulation size and placement over the pipes and around the pipes, and the effect of fins that could be attached to the pipes. The temperature distribution on the floor surface of a farrowing house can not be calculated analyt- ically. A numerical procedure must be used. The finite element method appears to be the powerful tool available to study the temperature distribution of such a complicate model. The general objective for this study was to calculate the temperature dis- tribution on the floor of a farrowing house for specified hot water pipe arrange- ments and insulation placement. Another objective was to develop configurations that will provide comfortable temperature distribution for both the sow and baby pigs. Specific objectives relative to the design of a new hot water heating system evolved after the first analysis was completed; these are discussed later. H 11-111... snlluhulluiul . _ H . Irllllll IIIlII .IIIIIJ w. IIIII L Figure 1.1 Pipe arrangement in a typical hot water heating system of the fallowing house which has 10 pens. Figure 1.2 Pipe arrangement in a modified hot water heating system. II. LITERATURE REVIEW 2.1 Farrowing House Butchbaker and Shanklin (1965) studied the temperature regulating mechanisms of young pigs in the test chamber using four different room tempera- tures. They found a single newborn pig cannot maintain homeothermic status without supplemental heat despite a well-deveIOped shiver mechanism. Karhnak and Aldrich (1971) measured the room temperature and floor temperature of a farrow-to—finish building equipped with an under floor heating system. The room temperature, measured using thermocouples, ranged from 16 ° C to 21 ° C (61 ° F to 69°F) in the winter time. The floor temperature ranged from 21°C to 39°C (69°F to 102°F). They observed that pigs usually laid across the front of the pen, although the warmest spots were along both sides of the pen beneath the guard rails. Spillman and Murphy (1976) found that producers with totally slot- ted floor creep areas tended to keep the room temperature around 27°C (80°F) while those with partially slotted or solid floor creep areas maintained room tem- peratures from 16°C to 24°C (60°F to 70°F). They observed that pigs more than 7 to 10 days old tend not to sleep under heat lamps. Muehling and Stanislaw (1979) provided the important design factors for farrowing units whose floors are solid or slotted. They suggested the room tem- 5 perature of solid and slotted floors to be 15.6° C to 18.3° C (60°F to 65 °F) and 21.1°C to 23.9‘C (70°F to 75°F), respectively. The floor temperature for a litter at farrowing was suggested to be in the range of 29.4 ° C to 32.2 ° C (85°F to 90 ° F) for the first three days of life while the comfortable floor temperature for a sow was 15.6° C to 18.3° C (60°F to 65°F). Van Fossen and Overhult (1980) provided the fundamental information to select, design, install and operate an electric or a hot water floor heating system. They emphasized that heating pipes across the sow area must be insulated with a 1.3 cm to 2.5 cm (0.5 inch to 1.0 inch) thickness of rigid, non-deteriorating insulation. They recommended the heated floor area of from birth to weaning as 0.56 to 1.4 m2 per litter (6 to 15 [£2 per litter). England at al. (1987) stated that the baby pig areas, on solid or slot- ted floors without bedding, should be kept at 32.3 ° C to 35.0° C (90°F to 95 °F) for the first few days, and then in the 21.2 ° C to 26.7°C (70°F to 80°F) range until weaning at three to six week of age. The ideal floor temperature distribution in the farrowing house can be achieved based upon the literature. The floor temperature in the sow area should be kept uniform through the whole sow area in the range of 15.6° C to 18.3° C (60 ° F to 65 ° F) regardless the change of the hot water temperature and the age of baby pigs. The floor temperature in the baby pig area, however, should be con- trolled according to the age and the weight of baby pigs. Furthermore, it will be more desirable the baby pig area provides several micro temperature environments which baby pigs can choose by themselves because the individuals vary in their preferred temperature. It is not economical to keep the whole litter area (1.95 m2, 21 ft?) warm in the temperature range of 29.4 ° C to 32.2 ° C (85 ° F to 90°F) for 6 the first few days of life (Muehling and Stanislaw, 1979). The small heated area (0.56 m2, 6 ft”), only about 30 % of total baby pig area, is necessary for the new born pigs (Van Fossen and Overhult, 1980). The ideal floor temperature distribu- tion in the baby pig area was assumed in this study as that of providing a uni- form temperature in the range of 29.4 ° C to 32.2 ° C (85 °F to 90 °F) on the over 30 % of total baby pig area. Moreover, it should provide several temperature ranges to satisfy the baby pigs individually. 2.2 Three-Dimensional Finite Element Analysis The finite element method was introduced at the mid 19508 as a method of analyzing structures with reinforcing coverings. The method has become a powerful computational tool in the field of structural mechanics, fluid mechanics, and heat transfer, especially for the analysis of irregularly-shaped objects having different materials or complex boundary conditions through extensive rearch. For the case of three-dimensional steady state heat transfer, the pro- cedure was completely described by Zienkiewicz et al. (1967). The transient heat conduction problem for two dimensions was performed by Wilson and Nickell (1966) using a variational principle. They also solved time dependent problems using a single step technique. A detailed description of the general theory of finite element method is given in several textbooks such as Zienkiewicz (1977), Segerlind (1984) and Allaire (1985). 7 The accuracy of the finite element method for steady state heat transfer was studied by Laura et al. (1974) who calculated the error between analytical and finite element results for two cases whose domains were extremely compli- cated. Good agreement, less than 1 % error, was obtained between the finite ele- ment results and the analytical solutions for a hexagonal model with a concentric, circular hole and a square model in a nonhomogeneous media. Since the preparation of numerous input data is necessary and is a tedi- ous task, automatic input data generation is nearly a necessity when solving a three-dimensional problem. Akyuz (1970) presented a scheme for generating input data in two- and three-dimensional space using the concept of natural coordinate systems. He divided the solution domain into subdomains depending on the field quantities and the complexity of the geometrical form. Cavendish et al. (1985) describe the algorithm for the computer generation of tetrahedral finite element meshes for solids. The proposed algorithm was separated into two independent modules. First, the node points were defined within and on the surface of the solid. Then, the node points were automatically connected to form well- proportioned tetrahedral finite elements. To minimize the memory space and computer time, a banded matrix solution technique is used. The matrix should have a bandwidth as small as possible. Grooms (1972) presented a simple and straightforward matrix bandwidth reduction procedure. The basic idea was to systematically move rows that are far apart and coupled closer together. He com- pared his method with other bandwidth reduction methods. Collins (1973) presented a method in which the engineer numbered the nodal points but the computer renumbered the nodes to minimize the bandwidth during calculations. 8 The computer restored the original numbering for output. 2.3 Finite Element Formulation The finite element method can be viewed as a numerical procedure for solving differential equations. The finite element analysis in conjunction with a variational principal is a powerful method for the determination of the tempera- ture distribution within a complex body that has difl'erent material properties, an irregular shape and mixed boundary conditions. The governing partial differential equation for steady state three- dimensional heat conduction (Kreith, 1965) is 6 6T 8 8T 6 8T 82( maz)+ayurw—ay)+—aZ(K,,,——az)+c2 o (21) with the boundary conditions T = TB on S, (2.2) and/or 8T 8T 6T K3613+Kw 8yI"+K“ le'+q+h(T 00) Con 32 (3) where T ( 'K) is a temperature that is a function of x, y, and z. X“, K", and K” (kW/ m 'K) are the thermal conductivities in the x, y, and z directions, Q (kW/m3) is an internal heat source or sink, q (kW/m2) is the heat flux over the surface, and h (kW/m2 'K) is the convection coefficient. Tc,o ( 'K) is the ambient temperature, and T3 ( 'K) is the known boundary temperature. The quantities of 9 1,, I, and I, are the direction cosines of a vector normal to the surface. 51 is the boundary surface where temperature is known, and $2 is the another surface where heat is gained or lost due to a convection heat transfer or a heat flux. The functional formulation, that is derived from the variational calculus (Pars, 1962), for (2.1) and its boundary conditions (2.2) and (2.3) is _ .1. 6_Tz a_z-. 6_T2_ 11.. V2[K”(az) +K"(ay) +K“(az) 2QT]dV (2.4) +fs[qT+—;-h(T—Tm)2]d5 Functional, II, must be minimized with respect to the set of nodal values {T}. The minimization of H occurs when where E is the total number of elements. arIM . . . . . 7 _8{T} 1n (2 5) IS given by Segerllnd (19 6) as The derivative 33;) =( Iv(‘)[B(e)]T [0(a)] [3(a)] dV + 1;?) h [M°)]T [N-(c)] d5 ) {T} (2.6) ‘ fva [NWT dV + [51m q [NWT dS — 155.)” Too [M617 (15 where [D(‘)] contains the thermal conductivities K... o 0 [0(8)] = 0 Kim 0 0 0 K3, 10 [NM] contains the shape functions, and [BM] is related to the derivatives of the shape functions. The set of integrals in (2.6) can be condensed by using the ele ment stiffness matrix [K M] and the element force vector {EFM} as aw) WT? = [KW] {T} — {EF‘e’} (2:!) where [KM] = fv(.)[B(‘)]T [0M] [BM] dV + [54,) [1 [NWT [M01115 (2.8) and {EF(‘)} = fva [M‘)]T dV — 491') q [M‘)]T (£5 + [54,) h T..[N("]T d5 (2.9) The final system of equations is obtained by substituting (2.7) into (2.5), giving an E , , _ a T} = gum )1{T}—{EF< 1}) —o (2.10) or [K] {T} = {EF} 1211) where E [K] = 21 [KM] E {EF} = z: {EF‘c’} e-l Before evaluating the element stiffness matrix [K(‘)] and the element force vector {EFM}, equation (2.9) and matrix [0“] can be simplified with the condi- tion of K” = Kw = K” = K“, Q = 0, and q = 0, because of the same thermal con- 11 ductivities in the x, y, and z direction, no internal heat source, and no heat flux over the surface for the models in this study. Therefore, the element force vector reduces to {1515(6)} = L50}. T..[N(‘)]T dS (2.12) while 1 o 0 [13(6)] =K, o 1 o (2.13) o o 1 Since the three—dimensional element used in this study was an eight node hexahedron, the matrices [NM] and [BM] are [M‘)]=[N1 N2 N3 N8] ' (9N, 8N2 8N3 6N8 ° 8:1: 6:: (9:: ° ' ' 6:: 6y 8y 6y 6y 8N1 (9N2 (9N3 . . . 8N8 82 82 82 02 The coordinate transformation, from the global to the natural coordinate system, allows the boundaries of elements to be distorted, and requires the integrals in equation (2.8) and (2.12) to be evaluated numerically using a Gauss - Legendre technique (Segerlind, 1976). The global coordinate system (x, y, z), the natural coordinate system (5, n, g), and the location of the eight - node are shown in Figure 2.1. The shape functions and their derivatives for the natural coordi- nate system are given in Table 2.1. 12 Figure 2.1 Location of eight nodes in natural and Cartesian coordinates. 13 Table 2.1 Shape functions and derivatives for eight node hexahedron. Node Shape functions al-E)(1-n)(l-§) gnarl—mus) -;—(1+e)(1+n)(1—c) g—(l—emmu—e 51—50-5119) gnarl-mum 511.511.5114..) 5141111517105) Derivatives 6N,- BN, (SN,- 66 817 a; = —%(l—n)(l—s') g—c-nXI—c) %(1+n)(1—§) --1-(1+n)(1-§) 8 --1-(1-n)(1+§) 8 51—554..) %(1+n)(1+s‘) --;-(1+n)(l+s) —%(l—€)(1-§) -%(1+€)(1-§) gown—c) gu-eu—g) -%(1—€)(1+§) gamut.) 514.515.) 1 ‘8—(1-5)(1+§) —%(1-€)(1-n) --513-(1+€)(1-n) —-:,—(1+€)(1+n) -%(1-€)(1+n) 51-90—11) %{I+€)(l—n) %(1+E)(l+n) l §(1—€)(1+17) 14 The change in the increment volume dV is dV = dz dy dz = Idet[J] I dEdndg '2222fi' 06 (95 35 8: 8y 82 J: ___ l 1 8n 8n 8n fififi .6: as 6s. where [J] is the Jacobian matrix of the transformation The Cartesian coordinates are given by I=2N1X. s-1 8 y=EN.Y. 1-1 s z==1Z3Pfl£L 5-1 'aN, 8N2 8N8 ° 66 66 e . . 66 [J] _ 6N, 8N2 .. 6N8 _ 81) 817 817 3N, 8N2 8N3 _ 6c 8s 6; and the limits of integration are from -1 to 1 for each coordinate variable. 'X. 1'12.1 X2 Y2 22 and each column of [B] is given by .X3 Y3 28 (2.14) (2.15) (2.15) where X,, Y,- and Z,- are the nodal coordinates. Substitution into (2.15) yields (2.17) 15 . 8N,- . . 8N,- . 82 8E 8N- 8N- 8(6) = __1 = J-1 -——‘ 2.18 [ (6.0s)] 8y I 1 an ( ) 8N,- 8N,- 32 ,5-13 , 8g j-l,8 The change of variable in the surface integral is dS=dz dy = Idct[J]IdEd17 (2.19) Since convection heat loss in this study occurs only on the top surface, 5’ = l, the Jacobian matrix for the surface integral becomes @2212 as as 65 8: 8y 82 J = . I]... {-1 an 77,—" 37 (2 20) _o o 1 The unit value assigned to the diagonal allows the inverse matrix of [J] to be evaluated. Substituting (2.14) through (2.20) into (2. 8) and (2. 12) gives [KM] = 11,1.le [B(°)(€,n,§)lT to“) [B“’(€.m§)l Idet [J] I dédnds (221) 1511,1111 [M‘)(€,n)]T [M‘)(€.n)] Idet [J] I dedn {we}: 1 1,1,1 h T..[N‘¢( (521)] Idetmldedn (2-22) The Gauss - Legendre quadrature was applied to numerically evaluate the integration shown in (2.21) and (2.22). Since the highest order of the polynomials that occur in [8(‘)]T[D(‘)][B(')] and [M‘)]T[M‘)] of equation (2.21) is two, the 16 number of integration points (11) becomes two for each coordinate direction. The sampling points of 49.577350 and a weight coefficient (H,,-,, and H,,-) of 1.00 were used to evaluate the integrals. Eight integration points were required for the volume integral and four integrating points were required to evaluate the surface integral in (2.21). Since the highest order of polynomials in [NM] is one, one integration point is required for each direction. The sampling point for one integral point in the Gauss - Legendre is 0 and the weight coefficient (H,,) is 2.0. One integration point is required in (2.22). The numerical integration changes (2.21) and (2.22) to the following final 2 2 2 [KM] = 2 2 2 [f1(€i17lj1§k)Hijk l ldetlJll (2-23) i-lj-lk-l + 223 223 [f2(€i:’7j)Hijl Idet[J]| i-lj—l where [1(68'1771' 1gb) = [B(e)(€i1nj1gk)]T[D(e)(€i:nj1§k)][B(e)(€i:7”:ng f2(§i,’7j) = h [Me)(§£:’7j)lT [Mdfinflfll Hijk = HUI = 1.0 and 1 1 {EN} = h T... 21.231113155175115 1 ldet [J] I (2.24) a- )- where fa(E.-,n,-)=[ 0 o o 0 i111. H,” =2 LLV 44 III. ANALYSIS OF A TYPICAL FARROWING HOUSE The first calculations performed were done on the typical pipe layout shown in Figure 1.1. The cross section of the concrete floor heated with hot water is shown in Figure 3.1. A two-dimensional analysis was performed. This assumed that the pipes extended an infinite distance parallel to the sow area. The pipes beneath the sow area were not include in this analysis. 3.1 Farrowing Crate Dimensions and Finite Element Model Most farrowing crates have dimensions of 152.4 cm (5 feet) wide by 213.4 cm (7 feet) long. The width includes an 45.7 cm (1.5 feet) young pig area on both sides of a 61.0 cm (2 feet) sow stall as shown in Figure 3.2. The repeated symmetry of farrowing crates reduces the region to be analyzed. The X - axis was defined as the direction of the alley, while the Y - axis was the perpendicular to the X - axis as shown in Figure 3.2. The Z - axis was defined as the direction of the floor depth, upward being positive. Boundaries with respect to the X - axis were the center of the crate and the right side of the crate, while boundaries with respect to the Y - axis were the center of the crate and the center of the alley. Each boundary was an axis of 17 18 [Floor . f _..4-.. - .v. .. 6.4 cm 'é- 10", -7.6cm' - ' " '.u' _‘.19cmdlapipa.- _- Figure 3.1 Typical cross - section of concrete floor heated with hot water. C v G . a A a 2.5cm-3.8cm ' , Rigidinsulation _. "4 . ' ' A [Plastic vapor. barrier fill sand. 10.2 cm 19 152.4 cm '75— ia l 213.4 cm e O a O O _—--—— . o O . I O . . ___-__- W '1-45.7cm eli— 61.0cm 45.7cm —)| 111 Figure 3.2 Typical crate dimensions. X 1 20 symmetry. The shaded part shown in Figure 1.2 was the region analyzed. Two simple models were introduced to decide the amount of floor depth to be analyzed. The rigid insulation used to insulate the concrete floor from the earth in the typical system of a concrete floor heated with hot water was located at the 10.2 cm (4 inches) depth as shown in Figure 3.1. The first model, therefore, had 10.2 cm depth, whose boundary was assumed to be totally insulated. Hart and Couvillion (1986) reported that water from wells deeper than 600 cm (20 feet) has a constant temperature year round of approximately 10°C (50°F). There- fore, the second model had 600 cm depth, whose boundary temperature was 10° C. That was a real situation even though the model had too many elements. The result showed the temperature on the floor did not change significantly with the model depth. The bottom of the concrete floor, therefore, was assumed to be a nonconducting insulated boundary; 10.2 cm (4 inches) was selected as the depth of the model. This simplification reduced the computer memory requirement and the running time. The boundaries with respect to the Z - axis were the floor and the top of the rigid insulation beneath the concrete. The size of the model to be analyzed was 7 6.2 cm X 167.6 cm X 10.2 cm (30 inches X 66 inches X 4 inches). The boundary conditions on the surfaces were T = 60 ° C on the surfaces of the pipes k-g—r = q + h(T—T°°) on the floor surface 11 gl- = 0 on the all of the other surfaces except the floor 11 The temperature on the outside surface of the pipe was assumed as same as the hot water temperature 60° C (140°F) regardless of the thickness and the 21 material property of pipe. The values of input data, such as the surface conduc- tance (h) of concrete and the thermal conductivity (k) of concrete, insulation, steel, and copper, came from the ASHRAE Handbook (1981), Meyer and Hansen (1980), and Kreith (1965). The surface convection coefficient on concrete at zero air speed was 11.35 W/m2 °K (0.0139 Btu/hr inch2 °F). Since Muehling and Stanislaw (1979) recom- mended rigid insulation board for perimeter insulation and for insulation under concrete floor, particularly in heated floors, the wood or cane fiberboard was selected and its thermal conductivity was 0.0577 W/m 'K (0.00278 Btu/hr inch F). The new plastic insulations, such as polystyrene and polyurethane, are also used for a rigid insulation. But, they were not considered in this study because of their low structural strength. The thermal conductivity of concrete was 1.8025 W/m ° K (0.08681 Btu/hr inch °F). Van Fossen and Overhults (1980) recommended 60°C (140°F) as the water temperature and Spillman and Murphy (1976) found that most of the far- rowing houses were operated with a room temperature from 16 ° C to 24 ° C (60 ° F to 75 °F). Therefore, I assumed the water temperature to be 60 ° C and the room temperature to be 15.6 ° C (60 ° F), the coolest condition. 3.2 Calculated Temperature Values The temperature distribution on the cross section A - A’ in the Figure 3.2 is shown in Figure 3.3. The shape of temperature distribution shown in Figure 22 35 304 8 g .1 *5 25 BABY PIG AREA sow AREA 8 E ,2 204 ................... +------ ---..--..- I I 15 ................... T ------------ I ------------- I. ------------------- 00 381 75.2 1143 152 4 X — Axis (cm) Figure 3.3 Temperature distribution on the floor of the typical farrowing house heated with hot water. 23 3.3 was the same shape of floor temperature measured by Karhnak and Aldrich (1971). The 34 % of the baby pig area had a temperature higher than the recom- mended temperature range of 29.4 ° C to 32.2 ° C (85 °F to 90°F). The overheated area wastes energy and increases the operating costs. The 45 % area had a temperature below 29.4 ° C which is too cold for new-born pigs. Only 23 % area was in the optimal temperature range for baby pigs. In the sow area, the 34 % was in the desired temperature range of 15.6° C to 18.3°C (60°F to 65°F). The temperature contours are shown in Figure 3.4. The contours are straight line because of the assumption made about the pipes in the two-dimensional analysis. Since the highest temperature in the creep area exceeded the desired values, the possibility of placing insulation over the pipes was analyzed. Applying the wood or fiber board insulation, 1.3 cm (0.5 inches) thick and 5.1 cm (2.0 inches) wide, above the pipe helps to remove the hot space in the baby pig area as shown in Figure 3.5, but it was not enough to provide the temperature distribu- tion desired. The temperature distribution was not uniform within the comfort- able temperature range for the baby pigs 24 213.4 160.1- I 2 ii -; 13 a Egg 1, ‘ :b A if a | :l 5. 3 105.7- I ). fl *5! :f a: " it 53.3- 03 F 0.0 38.1 76.2 1 1 4.3 1 52.4 X - Axle (an) Figure 3.4 Temperature contour on the floor of the typical farrowing house heated with hot water. 25 35 l l l- [Temp. range for baby pig | 30“.--- ............ 1L ......................... IL ............ :5 I l v I l g I I E 25-1 BABY PIG AREA : sow AREA : BABY PIG AREA 0 E l l ,2 I I ~ ------------------ 1 ---------------------- l}- ------------------ -1 l Temp. range for saw I ------------------- -1-----—--u.-------—---—-—-—-r----—-----------—-. 00 38.1 76.2 1143 1524 Figure 3.5 Temperature distribution on the floor when 1.3 cm thick and 5.1 cm wide flat insulation is applied over the hot water pipes. IV. CALCULATIONS RELATED TO THE DESIGN OF A HOT WATER HEATING SYSTEM The calculations in the previous chapter indicate that the typical hot water heating system does not produce the desired temperature distribution in either the baby pig creep area or the sow area. The number of elbows in this sys- ' tem also makes it undesirable because of the increased construction costs, increased pumping power required and the possibility of leaks. The primary objective of the rest of this study was to obtain some proto- type designs for hot water heating systems that use straight pipes and provide the desired temperature profiles by utilizing insulation at various locations in the floor. The temperature distribution on the floor surface was calculated using a three-dimensional finite element computer program. The thermal properties, boundary conditions, room temperature, and the water temperature used in the previous chapter were also used in the three-dimensional study. This chapter discusses the design parameters. Three prototype designs are presented and discussed in the next chapter. 26 27 4.1 Finite Element Grid Generation The preparation of numerous input data for a three-dimensional finite element study is a time consuming task and a major source of errors. Programs that automatically generate the element input data are recommended. The advantages of generating the input data from a minimum amount of information are a) the ease of changing the few parameters for different problems, b) reduction of the hand labor involved, and c) avoidance of the human error. The necessary input data for generating a three-dimensional grid are as follows: 1. Number of regions, number of boundary points and the minimum number of x, y, and z coordinates of boundary points that could describe the model. 2. Region connectivity data which show the connection to other regions. 3. Number of required subdivisions in f, 17, and g directions that could be changed with the shape of region and significance for the region. 4. An integer node number that defines the region. The procedure to reduce the matrix bandwidth is necessary for minimiz- ing the memory size and the running time of the computer. The computer run- ning time required to solve the matrix is proportional to the square of the bandwidth (Grooms, 1972). The node numbers from a grid generation program usually have large bandwidths because the numbering of nodes is sequential within a region, therefore, the elements on the boundary of the region have a big 28 difference between the largest and smallest node numbers. The way to obtain a small bandwidth is to renumber the nodes so the nodes in each element are as close as possible. Since many algorithms for reducing the bandwidth are available (Grooms, 1972), the method considered herein was to number the element nodes in a sequence that starts at 6 = —1, g = +1 and r) = —1 and proceeds to f = +1, 5‘ = —1 and 17 = +1 within a whole model. This simple renumbering system proved advantageous in analyzing the temperature distribu- tion even though it did not give the minimum bandwidth. The following basic procedures are performed in grid generation and bandwidth reduction for the three dimensional problem. 1. Minimum input data defining the model are read. 2. The nodes are numbered sequently from left to right (6 = —1 to f =,+1), from tOp to bottom (g = +1 to g = —1) and from front to rear (17 = —1 to n = +1) skipping all previously numbered nodes within a region. 3. All nodes on the boundaries are stored for skipping when considering regions that are adjacent to the stored boundary. 4. The whole node numbers are stored and changed to the new ones renum- bered for bandwidth reduction. The grid generation program written in FORTRAN language is shown in Appendix A. One of the three-dimensional grid models used in this study is shown in Figure 4.1. The length in the vertical direction (z—axis) has been expanded for the sake of clarity. 29 Apovcaaxo mm £335 ho commaofiav 8.85208 .8 :0582 23 was am 304:3 839 .80: 8.23 .«o _ocoE 2: .5.“ Sofia—m H4. charm 30 4.2 Calculations Related to the Design of a Piping System The temperature distribution on the floor of a farrowing house depends on the thickness, location, and width of insulation above the hot water pipe. Several cases for one pipe were analyzed to get a ’feel’ for the temperature values as they related to the different insulation conditions. The cross section of the model used in this analysis is shown in the Fig- ure 4.2. The parameters W, D, tF, and tP are the width of insulation, the depth of insulation from floor, the thickness of flat insulation, and the thickness of per- imeter insulation, respectively. The model assumed that two hot pipes were per- pendicular to the farrowing crate. Nodal points and nodal elements of this model were 812 and 552, respectively. The CPU time spent in the VAX/VMS computer sys- tem was 17 minutes. The three-dimensional finite element heat transfer program used is given in Appendix B. Laura et al. (1974) obtained less than 1 % error between the finite ele- ment results and the analytical solutions in the extremely complicated model. The error for the finite element heat transfer program used in this study was tested using a problem whose temperature could be calculated analytically. The model was divided into the same shape of elements as used in this study. The maximum error of 5.8 % based on the Fahrenheit unit was obtained. The main error came from the flat shape of elements due to the small thickness of model as shown in Figure 4.1. 31 )e—w—fiT tFI J— flat insulation tP @- rimeter neulatian 106.8 cm Figure 4.2 Variables for the test model. 32 4.2.1 Placement depth of flat insulation (D) The temperature distribution as a function of the placement depth of an flat insulation, 1.3 cm (0.5 inches) thick and 15.2 cm (6.0 inches) wide, over the not-insulated pipe is shown in Figure 4.3. The only effect was that the deeper insulation placement leveled out the high temperature zone. A greater depth may be desirable when structural integrity of the floor is considered. It is impractical to place the insulation only 1.3 cm below the surface. 4.2.2 Thickness of flat insulation (tF) Figure 4.4 shows the temperature distribution obtained by varying the thickness of the flat insulation whose width is 15.2 cm (6.0 inches), placed at a depth of 2.5 cm (1.0 inches), and without perimeter insulation on the pipe. A large temperature difference occurred just above the pipe but no significant change occurred in the region 17.6 cm from the pipe. The temperature on the floor just above the pipe was decreased 5 ° C when compared with the 0.6 cm (0.25 inches) thickness of insulation. The temperature drop just above the pipe increased with the thickness of insulation nonlinearly as shown in Figure 4.5. The straight line was forced using only data exclude zero point in order to get the rate of a tem- perature drop with respect to the insulation thickness. The 1.94 ° C / cm (8.89 °F / inch) rate was obtained. This information would be used to decide the thick- ness of insulation in the new models. 33 35 ’ l I e—e Ne ineulati 1H: 0 - 1.3 cm (0.511.) I I h—A 0 - 2.5 cm (1 .0111.) I 4.4 0 - 3.8 cm}(1.5in.) I i 30"“ ————— ‘1‘“_" “'1'“ ___...r. ““““““ A l l I 9 I I I 2 l m‘\ l 3 —L—- _____ __ I I K _— _______ E” i d I ‘8 I E I //I I \\\\\ I .2 I -/ I \ - I 4’ _(_ s 20.. _____ __________ w __ x I I ‘E ’ I I I - " I I I ‘ - I I I ‘5 r l T 0.0 26.7 53.4 80.1 , 105.8 X - Axis (cm) Figure 4.3 Efiect of D, the placement depth of flat insulation (tF = 1.3 cm, W = 15.2 cm). 34 35 ‘ l T e—a No ineulatio H 6" - 0.6 c (0.251...) I I A-A tF - 1.3 8 (0.51m) I e-o tF - 2.5 c (1.0in.) l 514 tF - 3.8 cn’l (1.5in.) I I m— —————— 1—“—‘ “T" ——-—r —————— | I l I +\ I V f K e I C, —..I..— .. I 3 25— —————— —I—— ———+—— —— —————— E I / ,.. -.... \\ I s I V ‘1" \ I '— / I \ 20 -———— ;-' é ————— 1- ————— V ————— ' \ a; | I I \ E- W | | I I I I ‘5 r I I I I 0.0 26.7 53.4 80. 106.8 X — Axis (cm) Figure 4.4 Effect of tF, the thickness of flat insulation (D = 2.5 cm, W = 15.2 cm). Temperature drop ('0) 35 15.0 ‘ — Y = 5.9 x043 (R¢=0.99) ---- Y = 3.86 + 1.94 X (RI-£0.99) ”.2 10.0.. ““““ x e 5.0-I o" .I 0.0 1.0 2.0 3.0 4.0 Thickness of flat Insulation (cm) Figure 4.5 Temperature drop according to the thickness of flat insulation. 36 4.2.3 Width offlat insulation (W) The changes in the maximum temperature with the width of flat insula- tion were analyzed in the case of two diflerent thickness of a insulation, tF = 1.3 cm (0.5 inches) and tF = 2.5 cm (1.0 inches). The placement depth of the insula- tion was kept as D = 2.5 cm. No insulation was around the pipe. Significant temperature drops occurred with changes in the width of insulation as shown in Figure 4.6 and 4.7. The temperature drop just above the pipe increased linearly as shown in Figure 4.8 and 4.9 for each different thickness of insulation. When the linear regression was applied to each case, the rate of a temperature drop for 1.3 cm and 2.5 cm thickness of insulation was 0.36 ° C / cm (1.64 °F / inch) and 0.43 ° C / cm (1.96 °F /inch), respectively. The coefficient of determination (R2) in the linear regression was 0.98 and 0.96, respectively. 4.2.4 Thickness of perimeter insulation (tP) The insulation wrapped around the pipe lowered the temperature significantly as shown in Figure 4.10. There was no flat insulation above the pipe. A thickness the 0.5 cm (0.2 inches) resulted in a 11.4°C decrease. The round insulation reduced the temperature through the whole region as well as leveled the temperature distribution. There was not a big temperature difference among the thickness of perimeter insulation. The maximum temperature diflerence between the various thickness of insulation was 2 ° C (3.6 ° F). 37 e—e N0 flat Insul on I H w a 5.1 c ( 2.00..) I HI w - 15.2 c ( 6.0in.) I .4 w = 25.4 c' (10.0In.) I 30* ------ -I---- Ir'x- ----I- ------ 3 I / \ I g 25__. ————— -I—— Ir——II---~.-\ -—I— —————— L. a : / / ',e-—eoI-oo—-e~‘ : E f I \\\ .2 I // I \\. I , / 4'— 2o_ _____ 'J ____________ h _____ // I I I \ , / I I I \. \ | I | I I I 15 l I I F l 0.0 26.7 53.4 80.1 106.8 X - Axis (cm) Figure 4.6 Effect of W, the width of flat insulation (tF = 1.3 cm, D = 2.5 cm). 38 3‘5 i I e—e No flat loan on H W - 5.1 c ( 2.01m) I I b—A W - 15.2 c ( 6.0in.) I ewe w - 25.4 CI (10.01n.) I : 3* ------ :Iz: ----- “I :6 I ,1 I \ I 43 25 ————— -I—— --— I-—— --I- ————— -1 E / \ I ,2 | / / ....... ’*_._\x\ | ‘5/ "I" \\ 20- ————— If" ————— -I|I- ————— . ————— I I I x 5‘ I I I “‘ = I I I ‘5 I I I r I 0.0 26.7 53.4 80.1 1 06.8 X - Axis (cm) Figure 4.7 Effect of W, the width of flat insulation (tF = 2.5 cm, D = 2.5 cm). 39 1 0.0 3,05 Y = 0.16 + 0.36 x (R’=0.98) 6.0 -1 4.0 - Temperature drop ('0) l 2.0 - °°° I I I I I T I r 0.0 5.0 10.0 15.0 20.0 25.0 Width of flat Insulation (cm) T Figure 4.8 Temperature drOp according to the width of flat insulation (tF = 1.3 cm, D = 2.5 cm). 40 12.0 d 0 10.0.1 Y = 0.60 + 0.43 X (R’=0.96) E 8.0- “ II 8 '0 2 6.0- a E d a E I! 4.0— 2.0- 0‘0 r I T I I I I I I 0.0 5.0 10.0 15.0 20.0 25.0 Width of flat insuIatien (cm) Figure 4.9 Temperature drop according to the width of flat insulation (tF = 2.5 cm, D = 2.5 cm). 41 35 * I I e—e N0 insulatia H tP - 0.5 0 (0.21m) I I k-A 11> - 0.8 c (0.3in.) I e-o tP - 1.0 c (0.4in.) I e-v tP - 1.5 mi (06111.) I I m— —————— ———— —T— ————r —————— I I I 6‘ I | I b I I 2 I I I g 25— —————— -I--—-- -—--—-—-+-—-—--- —--+- ~~~~~~ a I I I E .2 I I I 20- ————— ——-—— FAK_—_— k _____ ‘ I Aififi'fih“~§:§\ I ’ 4’41?" I “ 2%“; W ' I ‘ “PM I. I I . O O 26 7 53.4 80.1 106 8 X - Axis (cm) Figure 4.10 Effect of the thickness of perimeter insulation (No flat insulation). 42 4.2.5 Thickness offlat insulation above the perimeter insulated pipe A thickness of 1.3 cm (0.5 inches) of flat insulation over the round pipe insulation whose thickness was 1.0 cm lowered the top temperature 0.9°C as shown in Figure 4.11. No significant difference was shown between the thickness of the flat insulation through the whole area. To level the top temperature, 1.3 cm thickness of flat insulation over the insulated pipe was desirable. 4.2.6 Width offiat insulation above the perimeter insulated pipe The Figure 4.12 shows the change in the maximum temperature with width of the flat insulation on the perimeter insulated pipe. The thickness and placement depth of flat insulation were 1.3 cm and 2.5 cm, respectively. The thickness of perimeter insulation was 1.0 cm (0.4 inches). The wide insulation increased the area of maximum temperature, but lowered that temperature. No significant temperature difference was shown in the area 22.9 cm away from pipe. 4. 2. 7 Summary The influence of the flat insulation and the pipe perimeter insulation on the floor temperature is now understood. Some design ideas could be deduced from that understanding. The reasonable location of the flat insulation is 2.5 cm (1.0 inches) below the floor level. The 0.6 cm or 1.3 cm thick flat insulation is necessary to remove the hot space in the baby pig area. The 15.2 cm (6.0 inches) wide flat insulation 43 35 H No flat insuI tion I T A—A tF .. 1.3 c (0.5in.) I I one 6" a 2.5 c (1.0in.) I I v-v tF = 3.8 c I (1.5in.) : I 3° ““ ————— ”I ______ ‘I‘ —————— I" ______ I I I i? V I l I g I I I *g 25— —————— —+ —————— + —————— ’r— —————— g I I I g I I I I— I I I I I I 20— —————— —I —————— -I- —————— I— ————— a I ,..I.\ I M i . I5 I I r I 0.0 26.7 53.4 80.1 106.8 X - Axis (cm) Figure 4.11 Effect of the thickness of flat insulation on the perimeter insulated pipe (D = 2.5 cm, W = 15.2 cm, tP = 1.0 cm). 44 35 I I H No flat insulation EH: W - 5.1 cm ( 2.01n.) I I A—A w - 15.2 crp ( 6.0in.) I I s.» w a 25.4 crln (10.01n.) I : 30'-I ““““““ "It ~~~~~~ T —————— {-— ————— -—-I {3 I I I v I I I 8 I | I 43 25*- ————— -I —————— + —————— I- —————— a I '6 " I 20 -I I I 15 Figure 4.12 Efiect of the width of flat insulation on the perimeter insulated pipe (tF = 1.3 cm, D = 2.5 cm, tP = 1.0 cm). 45 could level out the high temperature zone in the creep area. The perimeter insulation around the pipe is necessary when the pipe goes beneath the sow area, and the 1.0 cm (0.4 inches) thickness of perimeter insulation is appropriate. Additionally, thin flat insulation could be insulated over the insu- lated pipe in the sow area to level out the high temperature and to drop a highest temperature a little. The two - pipe system failed to keep the creep area in the 29.4 ° C to 32.2 ° C (85 °F to 90 °F) temperature range ; only one-third part of the creep area had a temperature over 25°C (77°F) in the case of no insulation. Therefore a three - pipe system is needed. V. PROTOTYPE MODELS FOR HOT WATER HEATING SYSTEMS IN A FARROWING HOUSE 5.1 Three Heating Pipes without Fins The temperature distribution on the floor of a farrowing house heated with three hot water pipes was analyzed using the model shown in Figure 5.1. This model contains a quarter of a farrowing crate. The variables used to find the optimum temperature distribution on the farrowing floor were the length (L1), width (W1, W2) and the thickness (T1) of the flat insulation over the pipes and the length (L2) of the perimeter insulation around the pipes. The thickness of a perimeter insulation on the pipes and the space between pipes were selected as 1.0 cm (0.4 inches) and 53.3 cm (21.0 inches), respectively. The depth of the flat insulation was 2.5 cm (1.0 inches) below the floor surface. The model consisted of 1694 nodes and 1194 elements. It took 27 minutes of CPU time on the VAX/VMS computer system to solve the system of equations. 46 47 Z/‘M .am .255; 8&9 02:... go .3608 one 6.. 83a€a> a.» 23E I“ \\ Z/‘M .g I—w—I _I.._III._ 48 5.1.1 Thickness offlat insulation Temperature distributions for an insulation thicknesses of 0.64 cm (0.25 inches) and 1.3 cm (0.5 inches) were calculated under the condition of L1 = 50.8 cm (20.0 inches), W, = W2 = 15.2 cm (6.0 inches), and L2 = 7 6.2 cm (30.0 inches). See Figure 5.1 for an explanation of symbols. The left drawing of the Figure 5.2 shows the temperature distribution along the X - axis when Y = 106.7 cm. This location is just above the pipe. The left side of the vertical dotted line in the left drawing is the sow area, and the right side the litter area. The high and low horizontal dotted lines show the tem- perature range appropriate for the sow and litter, respectively. The right drawing of the Figure 5.2 represents the temperature distribution along the Y - axis when X = 76.2 cm. The two horizontal dotted lines define the range of suitable tempera- tures for the litter. The temperature distributions shown in Figure 5.2 are the highest temperature values with respect to the X and Y - axis in the entire farrow- ing area. The flat insulation made the temperature in the baby pig area produced a lower temperature than desired, but provided a temperature approaching the desired temperature range in the sow area. The maximum 1.5 ° C difference was obtained between the 0.6 cm and 1.3 cm thicknesses of a flat insulation in the baby pig area. Without the flat insulation, there was a hot spot higher than the maximum desirable temperature 322°C (90°F) in the baby pig area, and the minimum temperature in the sow area was 1.6°C higher than the maximum desirable temperature 18.3 ° C (65 ° F). 49 A 80 «.2. I." «a :5 «.3 H 6: "II v: :5 wdm fl... 5 v nose—$5 2% ho mBaon» 23 Mo Somm— Nd 9:.me Ass .3 I of Ass .3 I x 560 — odm min flow 0.0 ad a _, _ _ m. n. Iou row m. w d O m 7mm m. InN , u 6.. IIIIIIIIIIIIIIIIIII Ion Ion IIIIIIIIIIII defindv Ea n.— In... one .1 I I I I I «$00.8 80 n; In... «.5 Aénudv Ea 0.0 In» aid ficfiudv Eu ed Ifi 31a 5:25.... 6: oz To 8:252. 6: oz To an on (0 .) unweduae J. 50 5.1.2 Width offlat insulation The width of a flat insulation in the baby pig area (W2) was changed from 0.0 cm to 15.2 cm (6.0 inches) to determine the effect of the width of the insulation. The sow area was insulated with L1 = L2 = 50.8 cm, Wl = 15.2 cm size insulation in the all cases. The thickness of insulation (T1) was 1.3 cm (0.5 . inches). The 5.1 cm (2.0 inches) insulation width was better than other widths even though a low temperature zone for the baby pig existed between the hot water pipes as shown in Figure 5.3. The highest temperature would be increased when the thinner insulation (T1 = 0.64 cm) was applied. 5.1.3 The length of perimeter insulation The 38.1 cm (15 inches) and 50.8 cm (20 inches) length of the perimeter insulation were compared in the flat insulation condition of TI = 0.64 cm, Wl = 15.2 cm, W2 = 5.1 cm, and L2 = 50.8 cm. The 38.1 cm length of the perimeter insulation showed wider high temperature distribution than the 50.8 cm length of insulation in the sow area, since the 50.8 cm length of the perimeter insulation showed the wider low temperature distribution than the 38.1 cm length of insula- tion in the litter area as shown in the left side drawing of Figure 5.4. The 50.8 cm length of perimeter insulation was more desirable than 38.1 cm length. 51 Ass 3 .II. a. .so “.2 I _3 .so new I 3 II. .1533” 3. so as: a: .6 seem as 23E AEoV 3.2 I > 560 F odm n.nn EON 0.0 a a _ _ n. .. Ion \ I. x m \\1111/ \s .W 6‘ Id: \1 \\\ . \ II. \‘\ 1.. II. “N m. slits \ 21.11.51... . u \ / ) . , \ m I .l IIIIIIII II I IIIIIIIIII L 1 IIIIIIIIIII Agog Eu «.9 I .3 «.0 ésodv Ea fin I .3 film $6.8 :6 ad a .3 a... mm A 35.8 .5 «.9 1 .3 4.... god .8 3 .. .3 510 650.8 Eu ad I .3 ole on (o . ) OJI‘anOdwOl 52 5.00 p 0.0m . Ase Ed I E. .85 S I s3 .80 «.2 "II ~>> .Eo wdm H 5v aouoemaoa «o 4.65. 23 «0 team Wm 0.53m AEov 3.2 I > new. new ad a 9 Ion Ina Ion €556 58.8 1 3 .I 393 52.9. .. 3 I an (o .) eanwedwel Ass «3 I x 5.368 48.8 .. 3 .I 35.3 42.8 1 3 I mp I In N Ton on (a .) OJnuDJOdUJOl 53 5.1.4 Recommended model Of the various combinations considered, the most desirable model when using three pipes is shown in Figure 5.5. All of the flat insulation had ,0.64 cm (0.25 inches) of thickness. The space between the flat insulation in the litter area and the sow area was provided to permit heat flow to the litter area, and to widen the high temperature zone in the litter area. The two dotted lines shown in Fig- ure 5.5 are the borders between sow and litter areas. The temperature distribution on the quarter-floor is shown in Figure 5.6. The temperature was 0.6 ° C higher than suggested for the sow, but nearly con- stant across the whole sow area. If the thicker insulation is used throughout the sow area, the temperature would be decreased a little, but not significantly. A small portion of the litter area was in the desirable temperature zone. If the flat insulation in the litter area is removed, a higher temperature zone could be obtained. Figure 5.7 and Figure 5.8 show the temperature contour and the three- dimensional temperature distribution on the whole floor, respectively. As shown in Figure 5.7, six separate hot areas exist for litter while a nearly constant tem- perature distribution exists in the sow area. A cooler area for the litter occurs on the floor between the pipes. It was difficult to get the desirable temperature distribution in the litter area when the only three hot water pipes were used for heating the floor. The possibility of attaching fins to the pipes to get an additional conduction effect was proposed. The idea is analyzed in the following sections because additional pipes in the heating system complicate the construction, and increase the operating 54 I“ 152.4 HI x 2.102 _,I k—53.3—+| 213.4 P—--— --- 32 o.om an 3 55 TON I In N I O n on (o .) eanoaeduJel l 10 N (3 .) eanoJeduJel mm 56 ' 152.4 76.2 X - Axis (cm) 0.0 Figure 5.7 Temperature contour on the floor of the recommended model for three pipes without fin. Cam 9505 33 u 8.— 49 L0 a —QU0 8 365 88 «o nos—Scam. . cc Oh: So a —d:O «808 $700.— an. w.w o.— fim-h at» m. “ax/l :j, «E...» Mic. _____o_o_o «. U» ____._..u....m. es INMOWOWMWWWWWNHMONOOOO 0’ aj. .u?u»o.«..»uoo«.«.ou...«.u.$ Oo/ooo oooooooooboflueueuuueueufiifii . t . é....§%&m§mé % Saunxu on». too ~ -~n§n~u~muuwwufitz§ .43“; -u~u~u$~uyum~v5§ m»? ”wuwwwumwwevhwmwyflflfia Q -nnwnufiufibflbflflv Q» ..«¢~»~.wmm§wum»mwz”? “may ~ é :fiwwuaaawwwvz 35 0 -§ -~ 1 I / r59 .is“fiéuaafiaauwew? fig? ~S§a 22¢ --~«§Wwwm..wmwmyz ~ : 3:: S i 3.... f? L -“\\\_m.mu.”.wvwww§\NM... é? m m. . ~ \Mm.» a a : §§u¢$o¢o¢oo-$§ bun \ n o o \ 9 o .1 a ,fle......~§\ m ewe§ p. - g 58 costs, with no significant improvement in temperature distribution. 5.2 Three Heating Pipes with Steel Fins A steel fin would be attached to each of the pipes to widen the high tem- perature zone in the litter area. The fin would probably be welded under the pipes for ease of construction. The thermal conductivity of the steel fin was 37.49 W / m °K (1.805 Btu / hr inch °F). The dimensions of the fin and insulation are shown in Figure 5.9. The length and thickness of the perimeter insulation were set at 50.8 cm (20 inches) and 1.0 cm (0.4 inches), respectively. The spacing between pipes was 53.3 cm (21 inches). The finite element model consisted of 2100 nodes and 1566 elements and required 28 minutes of CPU time to solve using the VAX/VMS computer system. 5.2.1 Length of fin Five difierent fin lengths, 0 cm, 15.2 cm (6 inches), 22.9 cm (9 inches), 30.5 cm (12 inches), and 53.3 cm (21 inches), were analyzed for flat insulation dimensions of 15.2 cm wide, 76.2 cm (30 inches) long, and 1.3 cm (0.5 inches) thick. The width and thickness of the fin were 25.4 cm (10 inches) and 0.5 cm (0.2 inches), respectively. The curves in Figure 5.10 show that addition of the steel fin increases the litter area temperature more than 2.5 ° C. This increase occurs through the whole 59 .am 8:3 8&9 8...: go 368 3.. nomad—=3” was .5 Ho 83a€a> ad 23E \\\N \V\\ Tllnz Ans so H E. :5 v.3 H ES :8 «.4 H E. .80 «.3 .II E .80 «.2. H :v am 3QO .«o ~3ch 23 .«0 80mm ofim Rama AEoV «.5. I > #60 F 0.0m mnn Emu 0.0 \ \ ’r“ 335 so no... .. h on. IIIIIIIIIIIIIII 450$ V Eu Qua I .3 one ésoé v Eu «.0. ! .3 one a: .03. 505.3 0.... an (o .) mmmedwel AEov a.x< I x 4533 so and .. .5 s... 4:5.va Eu mdn I .3 one r llllll god v :8 3N .- .3 «5 god v .5 «.9 .. b a... c: .2... 305.; on. T IO N (o .) mmmedwel l o n mm 61 litter area. However, the length of the steel fin was not an important factor even though a longer fin raised the temperature between pipes. That was because the temperature on the steel fin dropped rapidly with respect to the fin length as demonstrated in Figure 5.11 which shows the temperature distribution at the same depth as the fin. The 22.4 cm (9 inches) length of steel fin was the most desirable when the thickness of flat insulation was reduced to 0.64 cm (0.25 inches) as shown in Figure 5.12. 5.2.2 Thickness of fin Two fin thickness values, 0.5 cm (0.2 inches) and 0.25 cm (0.1 inches) were analyzed using a length of 22.9 cm (9 inches) and a width of 15.2 cm (6 inches). The flat insulation was 0.64 cm (0.25 inches) thick. Because the thinner fin produced a 1.1 ° C lower maximum temperature than the thicker fin shown in Figure 5.13, the 0.5 cm thick fin was more desirable. 5. 2. 8 Recommended model The most desirable model of using three hot water pipes with steel fins is presented in Figure 5.14 based on the insulation information obtained in the pre- vious section, and fin information in this section. Most of the sow area was insu- lated with 1.3 cm (0.5 inches) thick flat insulation to decrease the temperature in this area. On the other hand, the litter area was insulated with 0.64 cm (0.25 inches) thick to enhance the temperature. The space between the flat insulation 62 65 H LF - 15.2 cm ( 6.01m) H LF - 22.9 cm ( 9.0m.) b-A LF - 30.5 cm (12.01n.) 55‘ 04 LF - 53.3 cm (21.01n.) s ”I ‘ L 45- I' ’ g {r Aff E / k\\\o—---o”// a ll \\ / o / / b——{ I- I/ l’ / x/ // 25fl //’//// / // , , / ‘5 I I I I 0.0 26.7 53.4- 80.1 Y - Axis (cm) Figure 5.11 Temperature change on the steel fin. 63 .:w .86 wac— Eo 9mm :0 nomad—5mg as: ._o $2535... 2: ._o fiesta afim 9:33 AEov 3.2. l > 5.8— odm 33 new ad a _ a a. Tom Ina Ton “.538 .5 Ed I : I- A6598 .5 on; .. F I on (3 .) mnuuedwel A58 52 I x «.2 odm You 06 _ 9 low In“ Ion 453.8 .5 Ed I : .I A.c_8.8 so on.— .. p I on (o .) unmedmel 64 .3: .83 go @5329 2: ho scuba «fin 95mg A58 «3 I > 3.8 2.2 I x Bo. Bow 3..” new ad ad I , _ n. or 18 low m. w w Ian M In“ a .8 Ion [on T's..- ||||||| IIIII-IIII-IIII'IIIIIl-l'll'lll rull"|""ll+llllll'llll 35.8 so mad I h I. $5.8 Eu mad I b I. 3N8 :8 86 I ..= l ASN.8 :5 and I b .l on on (3.) unmedwel 65 I: 152.4 :IIX II—10.2—+I .LT ”"- 94 4°? 53 ‘81 '1“ , T. ’2 ’0 .IL ca JO I .33 ME 2' T " 1. ,i I8 L T _i. T- Y Unitzcm Figure 5.14 Recommended model for three pipes with steel fins. 66 of the litter area and the sow area was same as in the case of pipes without fins. The 30.5 cm (12 inches) long fin was attached to the center pipe since shorter 22.9 cm (9 inches) long fin to the side two pipes. That could widen a comfortable tem- perature space for baby pigs and raise the temperature of cooler area between pipes in the litter area. Figure 5.15 shows the temperature distribution for the quarter part of the floor and Figure 5.16 and Figure 5.17 present the temperature contour and the three—dimensional temperature distribution of the whole farrowing floor, respec- tively. The temperature distribution in sow area was even and within the ade- quate temperature range except for the small center portion of the sow area. A large part of the litter area was within the desirable temperature range. Widen- ing the space between pipes along with extending the length of the fin would be helpful in removing the cold zone existing in the upper and lower litter areas. Using a fin which is high in the thermal conductivity such as a copper would be another approach. 5.3 Three Heating Pipes With Copper Fins A copper fin that has a high thermal conductivity was introduced to (widen the desirable temperature zone for the litter and to reduce the size of fin. The thermal conductivity of copper is 377.23 W / cm ° K (18.17 Btu / hr inch‘ F) which is 10 times higher than steel. The variables for the copper fin are the same as those shown in Figure 5.9 for the steel fin. The insulation constants were same 67 .95 $96 5:3 803 09:3 .50 .8008 00058808.. go .50: 23 :o 005053.80 25330800. 3.0 95w?” Ass .2 I > 5.00 — 0.00 n._nn th 0.0 l I In 0 N N (a .) mmmedwel I 0 In on A508 3.2 I x «.05 0.00 You 0.0 _. L 9 _l- lllllllll nu- lllllll I1 I In N (3 .) mnmedwel I S on 68 213.4 Y — Axle (cm) l l 0.0 38.1 76.2 1 14.3 152.4 X - Axis (cm) Figure 5.16 Temperature contour on the floor of the recommended model for three pipes with steel fins. J .0008 mug—wave 882 ho GO— . .35: £5 a . $6 ohaadhwaeflwemfii 890 8.2 True—2.0 a ho . .5. u . H m. ”Saw . E 0.0 0.0 .+ ___.._.._ .. r... a... ___...«mmmuvouefi 1:, .3 _nrrmmmmmouxueuunvooué ,7, . 8 2 fitmouuuooumououwwunééo a A no .5). R0 090.00. .0 000 00 I ’I’ O, 11 ~ a __.F.«oouuuuouououuouuouoououoouhfirfiwfl’i’a’z j 1 z ___moouuuuouuuunuuouuuumouuuwuuwwfifi a” i ,, 1 —_ _OOOHONNONOfluONOONO’ONONNNHNN“”NHOHNONNH/W/////’ 1 11 TO __....u.....»...éé¢ 0 5 s OONWOONNOOOO”OOOOO“QQOOOO’O “.0.“”0"”N“"N"“NWMW/fl// ”’ fl] o9 oo 9 Q...” Wuuuuwuwwwum... w. ”Miagfifiwfiew.. :3: 0 E: $ ---u~uvrvvuumwww aw»? ééaoozté $E~u$$ --V~Sawwuwuuuuuw199/ ,8 aa3..s»§.....~.§ e».....»..%m%w«u ,. 9 ’o .3883. $1: E“e§¢§§§wuawmwemtta”? ’2: . 88“....ze 8?..5 e Sismzt is. m .g fig.“wWe"..fi:§wfiunfiwfiRM“...w.-- ,,t d .uuuwgnuoouooogxfix§$~§-$§0 foo/0000 - fi? wuéwyo 9 - ~~§\ , on .mmwwwnu.«8mw«»»&ovw~w§\ 5m“... 9 \ oouwuwww»wwww~\\ 1 mm O\\n\\\ \\ ,, (O o ) aanegadmal 70 as in the case of the steel fin. The finite element model was also the same as that used for the steel fin. 5.3.1. Length offin The four cases of fin length, 0 cm, 15.2 cm (6 inches), 22.9 cm (9 inches), and 30.5 cm (12 inches), were analyzed. The width and thickness of the fin were 25.4 cm (10 inches) and 0.5 cm (0.2 inches), respectively. The 15.2 cm long fin increased the highest temperature by 4.5°C as shown in Figure 5.18. The temperature distribution varied with the length of fin very significantly. Figure 5.19 shows the temperature changes with respect to the Y - axis at the buried depth of the fin. The temperature on the copper fin itself did not vary significantly with the length because of the high thermal conduc- tivity. Therefore, the copper fin gave the same efiect as widening the hot water pipe. The 15.2 cm and 22.9 cm long fin showed good temperature distributions. The 15.2 cm long fin showed a better temperature distribution when thinner fiat insulation (T1 = 0.64 cm) was used over the pipe and fin as shown Figure 5.20. 5.3.2 Thickness offin The effect of the thickness of the copper fin was analyzed using 0.25 cm (0.1 inches) and 0.5 cm (0.2 inches) thickness values under while keeping length at 15.2 cm, the width at 15.2 cm, and the thickness of the flat insulation at 0.64 cm. No significant difference attributable to the thicknesses of fins was found as shown 71 5.00 P A83 u.x< I > 0.w0 Jun A80 0.0 H «.3. .8o v.3 H ES .8o «A H 3. .8o «.3 H E .80 «.05 H a :0 .5008 .«o 505— 23 .00 833 wfim 239m cc .3055 I 83.9 I .3. one anfiu I .3 «.3 E006». I .3 o... so 4' ss \ ""1ll' l/i.\<\)lo \0 / \ IO \ \ Ifllullnlllhaiilfi '''''''' I. AEOV 3.2 I x 3a.“: so 99.... I .3 o... 3o... 0 Eu Qua I .3 fa 866 0 so «.2 I .3 a... Inn :0 338 «305.3 I 72 65 H LF - 15.2 cm ( 6.0in.) H LF - 22.9 cm ( 9.0m.) h—A LF - 30.5 cm (12.0in. ” r ‘5 55- f \ \ I I \ I I, ’8 I L 45— I I _1 f e I f \ " / 3 I / ‘5 ~— 3 ’ / E 35-3 I l ,2 H // / / / 25— / / / / / :‘a/ $/ 15 * , l l 0.0 26.7 53(4 86.1 Length (cm) Figure 5.19 Temperature change on the copper fin. 106.8 73 5.00 — 0.00 80 .8008 88 3.2 I » 0.00 _ 5.0N 0.0 [I 8.8.8 .5 3... I = I. 38.8 E... 3.. I s I l 0 N (o .) summedwe J. 0. ON 00 muo— Eo 0.00 no 0052880 .80 00 88835 2: 00 88.0.0 00.0 0.8070 8.8 8.2 I x 0.. 8.88 so 3... I p .I 8.88 so 3.. I p l I If) N (0.) wmmodwu I 8 0n 74 5.00 p 80 .8008 .00 88585... 8.0... .00 880m 00...... 8.8th 88 82 I > 88 .3 .. x QB QB 68 c... «an 8.8 «.8 , od .. L . L L m. .. IIIIIIII IIIQIIIIIIII 0. Tom low m. w m T0N m. .I0N 0 0 T00 100 8.8 58...... I h: .I 8.8 588 I h .I 8N8 :3 .3 I ..... .l 888 50.8 I h I. an an (o .) mmmedwel 75 in Figure 5.21. The 0.25 cm thickness fin is the most desirable from the economi- cal viewpoint. 5.3.3 Width offiat insulation The influence of the width of the flat insulation over the finned pipe was analyzed for the 22.9 cm (9 inches) long copper fin as shown in Figure 5.22. The widths of insulation were 15.2 cm (6 inches), 22.9 cm (9 inches), and 30.5 cm (12 inches) under the condition of 22.9 cm long fin and 1.3 cm thick insulation. The temperature distribution was influenced significantly by the width of the insula- tion. The same width of insulation and the length of fin, the 22.9 cm wide insula- tion and 22.9 cm long fin, showed the best temperature distribution in Figure 5.22. 5. 3.4 Recommended model Figure 5.23 shows the desirable farrowing unit heated using three hot water pipes with the copper fins attached. The 22.9 cm (9 inches) long fin was attached to the center pipe since 15.2 cm (6 inches) long fin to the side two pipes to widen a comfortable temperature zone in the baby pig area. The thickness of flat insulation in the sow area was 1.3 cm (0.5 inches) while the thickness in the litter area was 0.64 cm (0.25 inches). The width of insulation and the length of fin was set using the result determined in Figure 5.22. The model gave a good temperature distribution as shown in Figures 5.24, 5.25, and 5.26. The tempera- quz 14".“ \ | - 76 5.00 .. .80 8008 0:0. 8o 0.00 :0 80.8.38. .80 ..o .30....» 8.... ..0 880m. 00.... 8.80.0 A88 82. I > 0.00 0.0 5.00 0.0 _ . _ n. . 100 T00 I00 8.3... .8 8n I s. fa 050.0 0 Eu 0.00 I .3 In! 1.00 85.9 8 so «A... I .3 To (3 .) unwedwel AEOV a.x< I x A 8.0.8. .5 8.. I s. a-.. 8.8 8 so «.8 I s. at. 050.0 V 80 0.0. I .3 I 1.00 [.00 (o .) emuuedmel 77 ,8 152.4 A k—‘OZ-d I" s a II-I—-——76.2——I—>l T ' 25.4% i ‘1 ‘2'? l 43': | *T I‘ T I e I | J. a ro J. V’ I0 0 :2 . :1 2‘5. I! laid “‘ . T Ii T I: I l J. . 3 .‘L. ""772 I! ‘4 . . T .L l T I i I I I I .L l 1 Y" I in.“ #13 I: 1M ‘ ' Unit : cm le——101.6 4.: Figure 5.23 Recommended model for three pipes with copper fin. 78 .80 .8008 0...? 80.0 8.8... .00 .8008 8888888.. .00 .000 8...... :0 80.802800 88.880880. v0.0 8.80.0 88 2.2 I > 58. 8.. n... 58 o... . . _ n. l O N I 0 0 (o . ) unwedwel I 8 00 0F 1.00 I In N (o .) unmedwel I 8 79 213.4 160.1 - 78.9 106.7 - Y - Axis (cm) 18.9 53.3 - 0.0 I 1 I 0.0 38.1 76.2 X - Axis (cm) Figure 5.25 Temperature contour on the floor of the recommended model for three pipes with copper fins. .95 L238 4:3 83a 3:: no.“ 358 wovcoafigfi go someontammv 238383 _d:o_m:o8%-8£fi Sum Rama ”NH. :EOV ”Taft; #3: : N.©°~ 7% j :3 mg. O ’00.. ONOMOWOK. / II 9N?! Q‘I’ IO QQQNMQQ§§§§§Q>6 _ - #150. --~§vw ’ . E 4%, .....a\\.\mu...”.. /’ ’OOOwOV/ .‘x’. O’NOO 0 OO O- h 5 OOOOVWN . .\\ (Q .) aan'eJadmal 81 ture distribution in the sow area was very similar to the case of steel fins, but a wider comfort temperature zones for the litter was obtained even though the size of copper fins was smaller than the steel fins. 5.4 Effect of Room and Hot Water Temperature The temperature distribution on the floor of a farrowing house is affected by the temperature of the hot water running through the pipes and by the room temperature. Generally, the floor temperature is controlled with the water tem- perature. Van Fossen and Overhults (1980) recommended 60° C (140°F) as the water temperature and not more than 5.6 ° C (10 ° F) as the temperature difference between the supply line and the return line to obtain a high thermal efficiency. Heat loss for four different water temperatures, 62.8 ° C (145 °F), 60.0 ° C (140°F), 572°C (135°F), and 54.4° C (130°F), were analyzed using the model consisting of three pipes and a steel fin (Figure 5.14). The water temperature had little efl'ect on the floor temperature in the sow area, but did significantly affect the temperature in the litter area as shown in Figure 5.27. The temperature in the litter area was raised by 2.0 ° C (3.6 ° F) in the litter area when the water tem- perature rose 5.6 ° C (10 ° F). The highest temperature in the litter area was some- what lower than the desirable temperature when the water was at 54.4 ° C. There- fore, the crate near the end of the pipe system could be cool for the baby pigs. The room temperature significantly influenced the floor temperature through the whole area as shown in Figure 5.28. The lowest temperature in the 82 .O . 93 9.3.9.383 Boo.— 2: 3 238380..— ..oaaB so; 2: mo accum— hmé «Ema Ass «3. I > A58 22 I x 5.09 ado n.n «new 0.0 «.2. 0.8 inn o.o _. _ u 79 _ L m. 0.9% I p .3 IIIIIIIIIIII .— IIIIIII I 0.98 I h To _ OoN.hn I .—. did _ o.+.+nI»ol IIIIIIIIIIII._. IIIIIII TON _ ION m. w w _ 18 m _ In“ a _ J _ b _ . IHIIII'IMJIII “\‘W sl I .l ._v lllllll 1 ..... .o. \. $.35 03.!» I » o1 I I I fir I can: 0.2... I .F 4.... 9735 03.8 I h To $.95 0.9% I » $0 on , on (0.) unwodwu 83 w... . as h hilzfuhv .0 . on 05000380... 00003 2: 0.0 0030000800 8000 2: 00 003E wad BREE AEOV n_x< I > sum or ode n.nn Adm od 0 _ h 0 mp 0.0.“. I p o-.. x. 0.0....— I » Ta . 0.....0. I p Ta .. 00 —.FN I. .P 0|. \~ g \I 10.... \\ \\\ \ InN .0. Fig l llllllll _ \ x \ / \ \ I|'l\bl}l‘)|l\¥ut‘l\l llllllllll AIIICI! \ / f‘\ N \ I x. \ IllIIi an (o .) unwodwol A58 2.2 I x .f x \ REE 0.0.3 I 0 ol IIIIIII .xIxQBmv 0.».9 I 0. 4.... rillullll\s< \ $.08 0.0.9 I » nIu \\ $.03 0.0.0. I p o-.. lie] I I In 0 N N (o .) mnmodwol I o n on 84 sow area was 1.9 ° C to 3.4 ° C (3.4 " F to 6.1 ° F) higher than the steady state room temperature. Raising the room temperature by 2.8° C (5°F) increased the tem- perature in the sow area by 2.5'C (4.5'F) which is around same as the room temperature change while the temperature in the litter area by 1.8'C (3.2'F). Spillman and Murphy (1976) surveyed most of the farrowing house operated with the room temperature from 16°C to 24°C (60°F to 75’F). Farrowing house with solid floor, typically concrete, tends to operate at somewhat lower tempera- ture. Figure 5.28 shows the same result. A solid floor with hot pipe heating sys- tem could be operated at a lower room temperature. Figure 5.29 gives the temperature distribution on the floor when the room temperature was 13.9 ° C (57 ° F) and the water temperature 62.8 ° C (145 °F). A desirable temperature distribution is obtained throughout in both the sow and the litter area. _.m 85 0030000800 000.03 E . m: V O . Que was 050209800 802 A h . um V O . 92 .«o ”85:58 E 003:3...va 003003808 omd 05mg 0:3 .3 I > A58 «3 I x 5.00. 08 an... new 0.0 0.0 . m, h _ up n p Iou Tom m. w « Ina m in“ m 0 a I I I I I I I IIIIIIIIIIIIIIIII Ion Ion on on (0.) mmwedwel VI. DISCUSSION AND SUNINIARY The floor temperature values for three new layouts of hot water floor heating pipes were calculated. The three layouts consisted of (1) pipes only, (2) pipes with steel fins, and (3) pipes with copper fins. The temperature distribu- tions on the floor for the three different layouts were similar. Each had six separate hot areas for the baby pigs per stall and a relatively uniform temperature distribution in the sow area. The six separate hot spaces in the litter area should encourage baby pigs to scatter, preventing piling up and crushing. Moreover, since the high temperature zones in the litter area were well out of the sow area, baby pigs might stay out of the sow area where there is the risk of being crushed by sow. These new models could reduce the baby pig loss because Liptrap et al. (1987) reported that one of the main causes of baby pig mortality was crushing and injury. The lower temperature zones between the pipes in the baby pig area would be used by the baby pigs that prefer a lower temperature environment because individual baby pigs vary in their preferred temperature. Several groups of litters with substantially different heat requirements owing to their age or weight are in a farrowing room at the same time. Each crate in the typical far- rowing house, however, has the same temperature environment since it has the same type of pipe circuit. Therefore, it is impossible to satisfy the needs each 86 87 litter in the typical farrowing house at the same time. Controlling the fin size and insulation could solve the decrease in water temperature problem in the typical farrowing house. Van Fossen and Overhults (1980) recommended that the water temperature should not drop more than 5.6 ° C (10°F) from the supply line to the return line. Such a difference of water temperature between the supply pipe and the return pipe makes the temperature on the litter area near the return pipe to decrease by 2 ° C (3.6 °F) compared with the area near the supply pipe as shown in Figure 5.27. If a larger fin and/or less insulation is used for the crate near the end of return line, the temperature drop in the litter area would be decreased and the temperature drop from inlet to outlet could be. tolerated. The new models have simple heating pipe circuits that reduce the pump- ing resistance by 36 % compared with the complicate heating pipe circuit of the typical layout shown in Figure 1.1. The comparison is given in Appendix C. The head loss due to the pipe fittings was significantly diminished in the new model. Therefore, that simple circuit would reduce the operation cost. The room temperature in the new model can be somewhat lowered than that in the current model while still keeping a desirable temperature in the sow area as shown in Figure 5.29. Some advantages of the new models are : 1. Preventing baby pigs from piling up on each other and being crushed by the sow owing to the desirable temperature distribution. In 88 Baby pigs can select their comfortable temperature area because various temperature zones exist within the litter area. Heating the only necessary area for baby pigs, not all of the baby pig area, can save the energy consumption. New model could remove the effect to the floor temperature change by the water temperature difference between the supply line and return line. Energy consumption and operation cost could be decreased by the lower room temperature and simple heating pipe circuit. VII. CONCLUSIONS The typical hot water floor heating system for a solid floor or partially slotted floor in the farrowing house has some problems in the pipe circuit and its thermal efficiency. To solve those problems, the new heating pipe circuit was introduced. To find the best model for the sow and her litter, the temperature distributions on the floors of various models were analyzed using the three- dimensional finite element method. The basic information about the heating ability of hot water pipes with flat and perimeter insulations and steel and copper fins attached to the pipes were obtained. This information was used to decide the numbers of pipes, the insula- tion size and placement and the fin size in order to obtain a desirable temperature distribution on the floor. Even though several models that have new pipe circuits were analyzed for the case of solid-floor farrowing system, the basic information could be helpful in the design of the new hot water pipe circuits for partially - slotted floor farrowing system, and a partially - slotted floor swine finishing pen. Three different cases, using only hot water pipes and using pipes with a steel fin or a copper fin attached were analyzed to determine the best model for each case. The three pipes across the crate were reasonable number of pipes and the fins attached to the pipes were necessary to widen the high temperature zone in the litter area. The perimeter insulation around the pipe running across the 89 90 sow area was essential. Three recommended models of each case showed adequate temperature distributions for the litter and the sow. The new models make it possible to build crates having different local floor temperature in the same far- rowing house. Energy consumption and operation cost can be reduced in the new pipe circuit. There will be some difficulties to attach fins to pipes and lay insula- tion over the pipes. The proposed models can be modified for manufacture if they are hard to construct as is. Future studies are suggested as follows : 1. All data showed in this study were derived from the simulated models. Measuring the actual temperature on the floor after constructing the recommended model is needed for validation of the simulation. 2. In the model, I was assumed there was no bedding on the floor and did not include heat produced by the sow. Further study is necessary to analyze the temperature distribution while including the effects of the sow. 3. The floor will be weakened compared with the typical floor when the wide fiat insulation is laid over the pipes. A stress analysis for the floor should be performed to determine its durability and strength. APPENDICES APPENDIX A GRID GENERATION PROGRAM C APPENDIX A PROGRAM GRID__3D C********************************************************************** C*** This program generates the node numbers and element numbers in 0*“ 3-D model and reoders the node numbers for minimizing the bandwidth C *********************III*********************************************** VARIABLES INBP : The number of boundary points INGR : The number of regions NBP : Boundary points NGR : Region number NBW : Band width N : The shape function XC,YC,ZC : x, y and z coordinates of the region nodes XP,YP,ZP : x, y and z coordinates of the boundary points JT : The region connectivity data NDN : Node numbers consisting of one region NN : The region node number NNRB : The node numbers on the boundary of the region XE,YE,ZE : x, y and z coordinates of the elements : Node numbers of elements XP,,YP ZP .NBP) ;(1.50) JT NBC, 6) (6,0 ,6)3 NN,,XC TC, 20. (ZETA) M(AX TA MAX, KSAI MAX), 10,10,10 NNRB :(NRG, 6, ZETA & ETA M’AX, KSAI & E’ A MAX ;(60,6,10,10) XE,YE,ZE,NE: (KSAIMAX * ETAMAX * ZETAMAX) ;(500) AXI 3000 ,3) NAXI : 3000) IELEOLD : 2000, ,8; IELENEW : 2000,8 COOGOO000000000000000000000000000000 91 C 92 DIMENSIONXP 150),YP 150),ZP(150),XRG(9),YRG(9),ZRG(9) DIMENSION Ng; ,NDN(8 ,NN 10,10,10 DIMENSION X 10,10,10 ,YC 10,10,10 ,ZC(10,10,10),NNRB(60,6,10,10) DIMENSION XE 500),YE 500),ZE(500),NR 8),NE(500),JT 60,6) DIMENSION (3000,3),NAXI(3ooo),IE OLD(2000,8), LENEW(2000,8) REAL N,KSAI DATA IN/5/,IO/6/,NBW/O/,NB/O/,NEL/O/,NODE /1 / C Input of the global coordinate and connectivity data 100 C 2 OPEN UNIT=IN,FILE=’GRIDIN.DAT’,STATUS=’OLD’) OPEN UNIT=IO,FILE=’GRIDIO.DAT’,STATUS=’NEW’) READ (IN,*) INRGJNBP DO 1001=1,INBP READ(IN,*) NBP,XP(I),YP(I),ZP(I) DO 2 I=1,INRG READ(IN,*) NRG,(JT(NRG,J),J=1,6) C***************************** 0*" Loop of generating elements C***************************** C DO 16 KK=1,INRG READ(IN,*) NRG,NKSAI,NETA,NZETA,(NDN(I),I=1,8) C Generation of the region nodal coordinates C DO 5 I=l,8 II=NDN(I XRG I = 11 YRS I =20”, CONTINfJE )CRGEQ§=XRG I) YRG 9 =YRG 1 ZRG 9 =ZRG(1) TR=NKSAI-1 DX=2. TR TR: TA-l DY=2./TR TR=NZETA-1 DZ=2./TR DO 12 I=1,NZETA TR=I-1 ZETA=1-TR*DZ DO 12 J=1,NETA TR=J-1 ETA=-1+TR*DY DO 12 K=1,NKSAI TR=K-1 KSAI=-1+TR*DX 93 0.125* 1.-KSAI)*(1.-ETA)*(1.-ZETA) .125* 1.+KSAI * 1.-ETA)*(l.-ZETA) .125* 1.+KSAI * l.+ETA)*(l.-ZETA) .125* 1.-KSAI * l.+ETA)*(1.-ZETA .125* l.-KSA1 * l.-ETA)*(1.+ZETA 125* 1. +KSA1 * 1 ..-ETA)*(1 +ZETA) .125* 1. +KSA1 * 1. +ETA)*(1. +ZETA) .125*1 L-KSAI)* (1. +ETA)* (1. +ZETA) KN 22222222 0040901000101— II II II II II II II ooooooo XC I,J,K’.—_-XO I,J,K +XRO L *N L YC IH,J,K =YC I, J, K +YRG L *N L =ZC( ,’J,K)+ZRO(L)*N(L) C Generation of the region node numbers C 56 45 46 KX1=1 KY1=1 KZ1=1 KX2=NKSAI KY2=NETA KZ2=NZETA DO 51 I=1, 6 NRT=JT( ’NRG,I) IF(NRT. EQ. o R. NRT. GT. NR0.) GO TO 51 DO 56 J=1, 6 IF(JT(NRT, J). EQ.NRG) NRTS=J IF(I.EQ.1 .OR. I.EQ.6) THEN L=NETA M=NKSAI ELSE LFSVEED‘TA .2 ..OR I..EQ 4) THEN M=NETA ELSE IF(I.EQ.3 .OR. I.EQ.5) THEN L=NZETA M=NKSAI END IF DO 60 KL=1,L DO 60 KM=1,M GO TO (45, 46, 47 ,48 49 50) I NN(NZETA, KL ,KM)=’NNRB(NRT, NRTS ,KL ,KM) KZ2=NZETA-l GO TO 60 NN(KL, KM ,1)=NNRB(NRT, NRTS ,KL,KM) KX1= GO T060 ‘ .....'J 94 47 NN(IQ.J,K1VI)=NNRB(NRT,NRTS,KL,Kl\/I) KY1=2 GO TO 60 48 NN(KL,KM,NKSAI)=NNRB(NRT,NRTS,KL,KIvI) KX2=NKSAL1 GO TO 60 49 NN(IG.,NETA,KIVI)=NNRB(NRT,NRTS,KL,KM) KY2=NETA-1 GO TO 60 50 NN(1,KL,K1\/I)=NNRB(NRT,NRTS,KL,IGVI) KZ1=2 60 CONTINUE 51 CONTINUE C IF 1.GT.KX2 GO TO 105 IF 1.GT.KY2) GO TO 105 IF KZl.GT.KZ2) GO TO 105 C DO 10 I=KZ1,KZ2 DO 10 J=KY1,KY2 DO 10 K=KX1,KX2 NB=NB+1 NN(I,.I,K)=NB 10 CONTINUE C g Storage of the boundary node numbers 105 D0 42 I=1,NETA DO 42 J=1,NKSAI NNRB NRG,1,I,J =NN NZETA,I,J) NNRB NRG,6,I,J =NN I,1,J) 42 CONTINUE DO 43 I=1,NZETA DO 43 J=1,NETA NNRB NRG,2,I,J = I,J,1) NNRB NRG,4,I,J I,J,NKSAI) 43 CONTINUE DO 44 I=1,NZETA DO 44 J=1,NKSAI NNRB NRG,3,I,.I =NN I,1,J) NNRB NRG,5,I,J =NN I,NETA,J) C 44 CONTINUE C Output of the region node numbers & x, y, z to AXI(NODE,3) C DO 63 I=1,NZETA DO 63 J=1,NETA DO 63 K=1,NKSAI E‘DEINN(I,J,K).LT.NODE)(GO TO 63 NODE,1 =XC I,J, AXI NODE,2 ==YC I,J,K AXI NODE,3 =ZC ,J,K N NODE =NO E NOD =NODE+1 95 63 CONTINUE C C Saving the elements and node numbers into ELEOLD(NEL,8) C L=1 DO 64 I=1,NZETA DO 64 J=1,NETA DO 64 K=1,NKSAI XE L =XC I,J,K YE; L =YC I, J ,K) (IA—N630 J ,K) L— 64 CONTINUE 1” C 0 i'-.“ DO 15 I=1,(NZETA-1) DO 15 J=2,NETA . DO 15 K=2, NKSAI :1 NR 1 =NKSAI*NETA*I+NKSAI* J-2 + K—l) NR 2 =NKSAI*NETA*I+NKSAI* J-2 + e NR 3 =NKSAI*NETA*I+NKSAI* J-l +K j NR 4 =NKSAI*NETA*I+NKSAI* J-l +(K-1) NR 5 =NKSAI*NETA* 1-1 +NKSAI* J-2 + K-l) NR 6 =NKSAI*NETA* 1-1 +NKSAI* J-2 + NR 7 =NKSAI*NETA* 1-1 +NKSAI* J-l +K NR 8 =NKSAI*NETA* 1-1 +NKSA1* J-l +(K-1) NE =NEL+1 DO 66 M=1, 8 66 [ELEOLD(NEL, M)=NE(NR(M)) C 15 CONTINUE C Output of last number of elements in each region C WRITE(IO, 300 NRG, NEL 300 FORMAT(3X, TNO. OF ELEMENTS IN ’13: REGION IS’,15) 16 CONTINUE C .C C C****1k*****It********************************************* C*** Reodering the node numbers for minimizing the band width C******************************************************** YzMAX- > MIN ZzMAX- > MIN XleN- > MAX NODE=NODE-1 LAST=NODE-1 DO 200 J=1,LAST Finding MIN X, MAX Y and MAX Z L=J 00000 000 96 JFIRST=J+1 DO 2101=JFIRST,NODE IF((AXI(L,2)-AXI(I,2)).GT.0.0001) THEN GO TO 210 ELSELI_F_((AXI (L,2)-AXI(I,2)).LT.-0.0001) THEN ELSE GIF((i§XIz(L6 ,,3)-AXI(I 3)). GT. 0. 0001) THEN ELSIi=I IF((AXI (L, 3)-AXI(I, 3)). LT .-0 0001) THEN ELSE 0111:sz (AXI(OL, l)-AXI(I,1)).LT..O 0001) THEN ELSE L=I END IF 210 CONTINUE C DO 220 M=1, 3 TEMP: 23%) AXIJ ,M =TE 220 CONTINUE NTEMP=NAXI L NAXI L =NAXI J NAXI J =NTE 200 CONT NUE G g Exchanging the node numbers DO 230 I=1,NEL DO 230 J=l,8 IELENEW(I,J)==O 230 CONTINUE C C CHANGE THE NODE NUMBER C DO 240 I-_—-1, NODE DO 250 —J=1,NI‘(IIL DO 250 K=1, 8 [FIE LEOLD(J, ,=K). .NE. M) GO TO 250 NEW( J ,)K 250 CONTINUE 240 CONTINUE C C***************************** C*** Calculating the band width C***************************** C DO 2801=1,NEL DO 280 L=1,7 DO 280 M=L+1, 8 LB=IABS( IELENEW I L)-IELENEW(I ,M))+1 IF(LB ..LE NBW) CO 0 280 280 C 97 NBW=LB NELBW=I CONTINUE C****************************************************** C*** Output Of x,y,z coordinates of nodes and node numbers C****************************************************** DO 2601=1,NODE WRITE(IO,261) I, AXI(I,J),J=1,3) FORMAT(4X,I4,5 ,3F10.5) CONTINUE DO 270 I=1,NEL WRITE(IO,271) I,(IELENEW(I,J),J=1,8) FORMAT(1X,915) CONTINUE WRITE(IO,71 NBw,NELBw FORMATéH ,1X,23H BAND WIDTH QUANTITY IS,I4, . CAL TED IN ELEMENT’,I4) CLOSEEUNIT=IN,STATUS=’SAVE’g CLOSE UNIT=IO,STATUS=’SAVE’ STOP END . ' i. ‘ APPENDIX B FINITE ELEMENT HEAT TRANSFER PROGRAM "‘ “-1—, nm‘ ' i‘cTaKh' “Bi—8A7: .: \ ‘ i < _ _ APPENDIX B PROGRAM I-IEAT_3D gill***********************1i!III*Ikill********************************** C*** This program determines the temperature distribution C*** on the floor of farrowing house by F. E. M. Input data C*** for this program comes from grid_generating program. C*************************************************************** C C ====== C VARIABLES C ====== C C NP : Total number of nodes C NE : Total number of elements C NBW : Band width C XE,YE,ZE : Coordinates Of elements C XG,YG,ZG : Coordinates Of global elements C PX,PY,PZ : Partial derivatives of shape function C X, Y, Z : Another expressions of Ksai, Eta and Zeta C NS : Element node numbers C ESM : Element stiffness matrix C EF : Element force vector C A : Column vector containg iT}, {F} and [K] C B : Derivatives Of the shape unctions C JGF : Last .pointer indicating the last storage for {T} C JGSM . " " F C JEND : N N H C K1 : Thermal conductivity of concrete C K2 : Thermal conductivity Of insulation C H : Convection coefficient C TCON : Thermal conductivity C TINF : Ambient temperature C TEMP : Initial steady state temperature C C ======== g SUBROUTINES C C BNDRYK : Assigning the thermal conductivity to the each element C NAT2D : Determination Of the calculating points in 2-D C NAT3D : " " in 3-D C DERSHP : Calculation Of the derivative matrix Of shape function ([B]) 98 OOOOOOOOOOOOOOOOOOO C C 10 99 ASMBL : Direct stiffness procedure SHAPE : Calculation Of partial derivatives of shape function MODIFY : Input Of the prescribed nodal values DCMPBD : Decomposition Of the grObal stiffness matrix SLVBD : Soving the system Of equations by backward substitution .XG,YG,ZG : (NP) .A JEND= NP+NP+BDW*NP) .NS NE)3 .SUBROUTI NDRYK & MODIFY .OPEN FILE CONVECTIVE SURFACE BOUNDARY CONDITION PWIKODNH ************************************************************** INIPLICIT REAL (A-H, O-Z) DIMENSION XG 14%YGQE7O, ,ZG(1470) COMMON/KY (g, ,ZE( 8) COMMON /NA U/ KSA’I( 8 A(8) ZETA 8) WC COMMON /AV A11907O ,,JGF JCSM, NP, NEW COMMON/ MéfK SM( 8, 8) ,EF 3(3) ,NS 1068' ,8) COMMON /SHA ,Y, z ,N(8)P (8),P (8), PZ(8), B(3, 8) REAL N, KSA/I ,,K1 OPEN UNIT=IN, FILE=’HEATIN. DAT’, STATUS=’OLV\) OPEN UNIT=IO FILE=’HEATIO .,DATY STATUS=’NE Y) DATA I /5/ ,IO/6/ DATA K1 /0 0868055/,K2/0. 0013888 / ,.H/O 013889 / TINF /60 /, TEMP /140 / DATA K3/ 180544 / READ(IN, *) NP ,NE,NBW WRITE(IO,*) NP,NE,NEW,K1,K2,K3,H,TINF,TEMP C Calculation Of pointers and initialization Of the column vector [A] JGF=NP JGSM=JGF+NP JEND=JGSM+NP*NBW DO 101=1,JEND A(I)=0.0 C Input of the node and element data (X,Y,Z & Node numbers) C 11 12 DO 11 I=1,NP READ(IN, *) II ,XC(I) ,YG(I),ZG(I) DO 12 I=1,NE READ(IN, *) II ,,(NS(I J), J=1 ,8) C*********************************1| 0*” Generation Of system of equations C********************************** C 100 D0 30 KK=1,NE C C Initialization of the element stiffness matrix and element force vector C DO 13 I=1 8 EF(I)=0.0 DO 13 J=1,8 ESM(I,J)=0.0 13 CONTINUE C C Retrieval of element nodal coordinates and node numbers C DO 14 I=1,8 J=NS(KK,I) XE I =XG J YE I =YC J ZE =ZG(J) 14 CONT I C g Check whether element has boundary convective surface or not ICON=0 DO 15 I=1,8 IF(ABS(ZE(I)-4.0) .GT. 0.00001) GO TO 15 ICON=1 15 CONTINUE C CALL BNDRYK(KK,K1,K2,K3,TCON) C Calculation of [B]T[D] [B] C WC=1.0 CALL NAT3D DO 17 K=1,8 X=KSAI K) Y=ETA( ) Z=ZETA K) CALL DE SHP(DET,ICON) DO 16 I=1,8 DO 16 J=1,8 DO 16 L=1,3 ESM(I,J)=ESM(I,J)+TCON*DET*WC*B(L,I)*B(L,J) 16 CONTINUE 17 CONTINUE C C Check of the boundary condition C IF(ICON .NE. 1) GO TO 25 CALL NAT2D DO 20 K=1,4 X=KSAI K) =ETA( ) Z=1.0 CALL DERSHP(DET,ICON) DO 19 I=1,8 18 19 20 25 30 C 101 D0 18 J=1,8 ESM I,J)=ESM(I,J +H*WC*DET*N(I)*N(J) EF I)=E (I)+H*WC*D T*TINF*N(I) CO NUE CALL ASMBL(KK) CONTINUE C*************************************************** C*** End Of the loop of generating the system of equations C*************************************************** C C CALL MODIFY(TEMP) CALL DCMPBD CALL SLVBD C Output of surface temperature 40 50 6O 70 WRITE(IO,40) FORMATUQgXJXEU)’,5X,’YE(I)’,5X,’ZX(I)’,5X,’TEMP’,//) DO 601=1, WS(2C(I)4.O).CT.0.00001 GO TO 60 ITE(IO,50)XG(I),YG(I),Z (I),A(I) FORMAT(3X,4F10.4) CONTINUE DO 70 I=1,NP $§ABS(ZG(I)—O.7348 .CT.0.00001) GO TO 70 ITE(IO,50) XG(I ,YG(I),ZG(I),A(I) CONTINUE CLOSE UNIT=IN,STATUS=’SAVE’ CLOSE UNIT=IO,STATUS=’SAVE’ STOP END C SUBROUTINE BNDRYK INIPLICIT REAL A-H COMMON REAL K1, 102 OgCK,K1,K2,K3,TCON) (8) C38(8) ZE(8) C********************************************* C*** Subroutine of assigning the K to each element CIIIIt******************************************* C MNwMMNMMMMNMNMNMMNNMMNMNNNMNNMMM wwwwwww 3:331 31ml? Z=(2E( (1 +ZE§I§)2 3’3: ABSX C 0.0).AND. (ABS(X L.T.150) ..AND ABS Y) CT 25. 95)..AND (ABS ).LT. 66. 0)AND. ABSZ. CT.2..2)AND (ABS(Z .LT.2.5)).OR. (ABS ...GT15 0)AND. (ABSOC) LT 20. 0)AND. ABSY ..GT 30. 28).AND. ABS ).LT 48.62) AND ABS ?.GT. 2.2) .AND.( S(Z .LT25)) .OR (ABS GT. 15.0).AND. (ABS ).LT 20.0) .AND ABSY G.T. 56.98).AND. ABS )L.T 66 mm ABsz C.T..22)AND( S(Z L.T..25)) .O.R (ABS .....CT200)AND (ABS ....LT300)AND ABSY CT. 34. 6o) .AND(.A(BAB )..LT 44. 45 AND. ABS 2 ..CT 2.2)AND sz L..T25)) .O (ABS .GT. 2o..o)AND. ((ABS ..LT 30.0)AND. ABSY .GT. 61.15).AND. AB )..LT 66 ..0)AND ABSZ ....GT22).AND S(Z ...LT25)) .O.R (KK..CE 25) AND. LE .27) ..OR KK.CE.55 AND. LE 57 .OR. KK.CE.61 AND. KKLE.63 .OR. KK.CE.73 AND. KK.LE.75 .OR. KK.CE.79 AND. KK.LE.81 .OR. KK.CE.67 AND. KK.LE.69 .OR. KK.CE.91 AND. KK.LE.93 .OR. KK.CE.451 AND. KKLE.453 .OR. KK.CE.457 AND. KK.LE.459 .OR. KK.CE.463 AND. KKLE.465 .OR. KK.CE.469 AND. KKLE.471 .OR. KK.CE.505 AND. KKLE.507 .OR. KK.CE.535 AND. KK.LE.537 .OR. KK.CE.541 AND. KK.LE.543 .OR. KK.CE.553 .AND. KKLE.555 .OR. KK.CE.559 AND. KK.LE.561 OR. KK.CE.547 AND. KK.LE.549 OR. KK.CE.571 AND. KKLE.573 ) THEN TCON= ELSE IF((3(1(ZI